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2011

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Difference approximation for an amphibian juvenile-adult dispersal mode
Azmy S. Ackleh, Keng Deng and Qihua Huang
2011, 2011(Special): 1-12 doi: 10.3934/proc.2011.2011.1 +[Abstract](317) +[PDF](140.6KB)
Abstract:
We consider an amphibian juvenile-adult population dispersing between ponds. We assume that juveniles (tadpoles) are structured by age and adults (frogs) are structured by size. This leads to a system of first order nonlocal hyperbolic equations. A finite difference approximation to this system is developed. Existence-uniqueness of the weak solution to the system is established and convergence of the finite difference approximation to the unique solution is proved.
Compartmental disease transmission models for smallpox
Burcu Adivar and Ebru Selin Selen
2011, 2011(Special): 13-21 doi: 10.3934/proc.2011.2011.13 +[Abstract](355) +[PDF](390.2KB)
Abstract:
Mathematical models are importants tool to develop emergency response plans and mitigating strategies for an anticipated epidemic or pandemic attack. With the objective of minimizing total number of deaths, total number of infected people, and total health care costs, mitigation decisions may include early response, vaccination, hospitalization, quarantine or isolation. In this study, smallpox disease transmission is analyzed with a compartmental model consisting of the following disease stages: susceptible, exposed, infectious, quarantined and recovered. Considering natural birth and death rates in a population, two models are developed to study the control and intervention of smallpox under the assumption of imperfect quarantine. First model is an ordinary di erential equation (ODE) model assuming exponential distribution function and the latter is a general integral equation model with gamma distribution. Numerical results are provided and model results are compared to demonstrate the di erences in the reproduction numbers, which can be described as the threshold for stability of a disease-free equilibrium related to the peak and nal size of an epidemic.
Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian
Goro Akagi and Kei Matsuura
2011, 2011(Special): 22-31 doi: 10.3934/proc.2011.2011.22 +[Abstract](322) +[PDF](127.9KB)
Abstract:
This paper is concerned with the initial-boundary value problem for a nonlinear parabolic equation involving the so-called $p(x)$-Laplacian. A subdifferential approach is employed to obtain a well-posedness result as well as to investigate large-time behaviors of solutions.
A model for stem cell population dynamics with regulated maturation delay
Tomas Alarcon, Philipp Getto, Anna Marciniak-Czochra and Maria dM Vivanco
2011, 2011(Special): 32-43 doi: 10.3934/proc.2011.2011.32 +[Abstract](297) +[PDF](477.0KB)
Abstract:
We develop a structured population model for the maturation process of stem cells in the form of a state-dependent delay di erential equation. Moreover, results on existence, uniqueness and positivity of solutions as well as conditions of existence for equilibria and representations of these are established. We give biological interpretations for the conditions of existence of equilibria.
On the Glimm Functional for general hyperbolic systems
Fabio Ancona and Andrea Marson
2011, 2011(Special): 44-53 doi: 10.3934/proc.2011.2011.44 +[Abstract](245) +[PDF](170.6KB)
Abstract:
In this note we discuss the features of a quadratic interaction potential for first order hyperbolic systems in one space dimension, aiming to suggest the need of defining a non local functional depending on the global wave structure of the solution in order to obtain a quadratic Glimm type functional for general hyperbolic systems.
From local to nonlocal in a diffusion model
Armel Ovono Andami
2011, 2011(Special): 54-60 doi: 10.3934/proc.2011.2011.54 +[Abstract](280) +[PDF](278.0KB)
Abstract:
In this paper we investigate the behaviour of a di usion equation where di usion depends to nonlocal terms. In a radial setting, by regarding bifurcation theory, we prove the existence of local branches of solutions, among them one is the global branch of solutions without bifurcation point.
Symmetry breaking and other features for Eigenvalue problems
Claudia Anedda and Giovanni Porru
2011, 2011(Special): 61-70 doi: 10.3934/proc.2011.2011.61 +[Abstract](337) +[PDF](341.5KB)
Abstract:
In the rst part of this paper we discuss a minimization problem where symmetry breaking arise. Consider the principal eigenvalue for the problem -$\Deltau = \lambdaxFu$ in the ball $B_(a+2) \subset\mathbb{R}^N$, where $N\>= 2$ and $F$ varies in the annulus $B_(a+2)\\B_a$, keeping a xed measure. We prove that, if a is large enough, the minimum of the corresponding principal eigenvalue is attained in a subset which is not symmetric. In the second part of the paper we show that the principal eigenvalue of an equation with an inde nite weight in a general bounded domain $\Omega$ can be approximated by a functional related to the energy integral of a suitable nonlinear equation; furthermore, we show that a solution of this nonlinear equation approximates, in the $H^1(\Omega)$ norm, the principal eigenfunction of our problem.
Coupled cell networks: Hopf bifurcation and interior symmetry
Fernando Antoneli, Ana Paula S. Dias and Rui Paiva
2011, 2011(Special): 71-78 doi: 10.3934/proc.2011.2011.71 +[Abstract](342) +[PDF](144.0KB)
Abstract:
We consider interior symmetric coupled cell networks where a group of permutations of a subset of cells partially preserves the network structure. In this setup, the full analogue of the Equivariant Hopf Theorem for networks with symmetries was obtained by Antoneli, Dias and Paiva (Hopf bifurcation in coupled cell networks with interior symmetries, SIAM J. Appl. Dynam. Sys. 7 (2008) 220–248). In this work we present an alternative proof of this result using center manifold reduction.
Characterisation of the asymptotic behaviour of scalar linear differential equations with respect to a fading stochastic perturbation
John A. D. Appleby, Jian Cheng and Alexandra Rodkina
2011, 2011(Special): 79-90 doi: 10.3934/proc.2011.2011.79 +[Abstract](307) +[PDF](345.6KB)
Abstract:
In this paper, we characterise the global stability, boundedness, and unboundedness of solutions of a scalar linear stochastic di erential equation, where the di usion coecient is independent of the state. The di erential equation is a perturbed version of a linear deterministic equation with a globally stable equilibrium at zero. We give conditions on the rate of decay of the noise intensity under which all solutions either tend to the equilibrium, are bounded but tend to zero with probability zero, or are unbounded on the real line. We also show that no other types of asymptotic behaviour are possible.
Exponential growth in the solution of an affine stochastic differential equation with an average functional and financial market bubbles
John A. D. Appleby and John A. Daniels
2011, 2011(Special): 91-101 doi: 10.3934/proc.2011.2011.91 +[Abstract](357) +[PDF](336.6KB)
Abstract:
We consider an ane stochastic di erential equation with an av- erage functional. In our main results, we determine the asymptotic rate of growth of the mean of the solution of this functional di erential equation, and show that the solution of the equation has the same asymptotic rate of growth, almost surely.
Explicit transport semigroup associated to abstract boundary conditions
Luisa Arlotti
2011, 2011(Special): 102-111 doi: 10.3934/proc.2011.2011.102 +[Abstract](255) +[PDF](327.2KB)
Abstract:
The linear transport operator associated with abstract bounded boundary conditions, $T_H$, is considered. It is shown that, in some particular cases, a convergent series similar to Dyson-Phillips series can be defined. Sufficient conditions assuring that the sum of this series is a C$_o$-semigroup generated by $T_H$ itself are given.
A remark on multiplicity of positive solutions for a class of quasilinear elliptic systems
Lynnyngs Kelly Arruda, Francisco Odair de Paiva and Ilma Marques
2011, 2011(Special): 112-116 doi: 10.3934/proc.2011.2011.112 +[Abstract](269) +[PDF](293.5KB)
Abstract:
Using variational methods, we prove a nonexistence and multiplicity result of positive solutions for a class of elliptic systems involving a parameter.
Existence results to a quasilinear and singular parabolic equation
Mehdi Badra, Kaushik Bal and Jacques Giacomoni
2011, 2011(Special): 117-125 doi: 10.3934/proc.2011.2011.117 +[Abstract](333) +[PDF](369.1KB)
Abstract:
We investigate the following quasilinear parabolic and singular equation, $ u_t-\Delta_p u &=\frac{1}{u^\delta}+f(t,x)\;\text{ in }\,Q_T=(0,T]\times\Omega\\ u>0 \text{ in }\, Q_T\; , u &=0\,\text{ on} \;\Gamma=[0,T]\times\partial\Omega,\\ u(0,x) &=u_0(x)\;\text{ in }\Omega $ where $\Omega$ is an open bounded domain with smooth boundary in ${\rm R}^N$, $1 < p< \infty$ and $0<\delta$, $T>0$, $f\in L^\infty(Q_T)$ and $u_0\in L^\infty(\Omega)\cap W^{1,p}_0(\Omega)$. For any $\delta\in (0,2+\frac{1}{p-1})$, $u_0$ satisfying a cone condition defined below and any $T>0$, In this paper we prove the existence and the uniqueness of a weak solution $u$ to $({\rm P_t})$.
Blow-up of solutions to some coagulation and fragmentation equations with growth
Jacek Banasiak
2011, 2011(Special): 126-134 doi: 10.3934/proc.2011.2011.126 +[Abstract](280) +[PDF](291.6KB)
Abstract:
The aim of this note is to show that solutions to a coagulation- fragmentation equation with growth can blow up in a nite time.
Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains
Michal Beneš
2011, 2011(Special): 135-144 doi: 10.3934/proc.2011.2011.135 +[Abstract](288) +[PDF](377.0KB)
Abstract:
We study a mixed initial{boundary value problem for the Navier{ Stokes equations, where the Dirichlet, Neumann and slip boundary conditions are prescribed on the faces of a three-dimensional polyhedral domain. We prove the existence, uniqueness and smoothness of the solution on a time interval (0, $T$*), where 0 $< T$* $<= T$.
Drawing the output of dynamical systems by juxtaposing local outputs
Farida Benmakrouha, Christiane Hespel and Edouard Monnier
2011, 2011(Special): 145-154 doi: 10.3934/proc.2011.2011.145 +[Abstract](328) +[PDF](473.6KB)
Abstract:
A dynamical system being described by its state equations and its initial state, we develop a method for drawing its output: It is based on the juxtaposition of local approximating outputs on successive time intervals $[t_i, t_(i+1)]0<=i<=n-1$. It consists in computing an approximated value of the state at initial point $t_i$ and an approximated output $y_i(t)$ on $[t_i, t_(i+1)]0<=i<=n-1$. An expression of the generating series $G_(q_r,t)$ for every component $q_r$ of the state $q$, an expression of the generating series $G_(y,t)$ of the output $y$ truncated at order $k$ are calculated and specified at every initial point $t_i$. We obtain an approximated output $y(t)$ at order $k$ in every interval $[t_i, t_(i+1)]0<=i<=n-1$. This method presents some theoretical advantages over Runge-Kutta methods: genericity, independency of the system and of the input, estimate of the error. So, an estimate of the suitable largest step can be computed. We have developed a Maple package for the creation of the generic expression of $G_(q_r,t),G_(y,t)$ and $y(t)$ at order $k$ and for the drawing of the local curves on every interval $[t_i, t_(i+1)]0<=i<=n-1$. For stable systems with oscillating output, for unstable systems near instability points, our method provides an appropriate result when a Runge-Kutta method is not suitable.
Time-like surfaces of prescribed anisotropic mean curvature in Minkowski space
Matthias Bergner and Lars Schäfer
2011, 2011(Special): 155-162 doi: 10.3934/proc.2011.2011.155 +[Abstract](246) +[PDF](305.9KB)
Abstract:
Solving an initial value problem for a hyperbolic system, we prove existence and uniqueness of time-like immersions of prescribed anisotropic mean curvature into Minkowski space $\mathbb{R}^(2,1)$ subject to geometric initial conditions.
Sharp pathwise asymptotic stability criteria for planar systems of linear stochastic difference equations
Gregory Berkolaiko, Cónall Kelly and Alexandra Rodkina
2011, 2011(Special): 163-173 doi: 10.3934/proc.2011.2011.163 +[Abstract](262) +[PDF](332.8KB)
Abstract:
We consider the a.s. asymptotic stability of the equilibrium solution of a system of two linear stochastic di erence equations with a parameter $h > 0$. These equations can be viewed as the Euler-Maruyama discretisation of a particular system of stochastic di erential equations. However we only require that the tails of the distributions of the perturbing random variables decay quicker than certain polynomials. We use a version of the discrete Itô formula, and martingale convergence techniques, to derive sharp conditions on the system parameters for global a.s. asymptotic stability and instability when $h$ is small.
A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case
Md. Haider Ali Biswas and Maria do Rosário de Pinho
2011, 2011(Special): 174-183 doi: 10.3934/proc.2011.2011.174 +[Abstract](352) +[PDF](330.1KB)
Abstract:
Here we derive a nonsmooth maximum principle for optimal control problems with both state and mixed constraints. Crucial to our development is a convexity assumption on the "velocity set". The approach consists of applying known penalization techniques for state constraints together with recent results for mixed constrained problems.
Existence for the linearization of a steady state fluid/nonlinear elasticity interaction
Lorena Bociu and Jean-Paul Zolésio
2011, 2011(Special): 184-197 doi: 10.3934/proc.2011.2011.184 +[Abstract](393) +[PDF](341.1KB)
Abstract:
A linearized steady state three-dimensional fluid-structure interaction is considered and its solvability is studied. The linearization (obtained in a previous work by these authors) that we deal with has new features, including the presence of the curvature terms on the common interface. These new extra terms, coming from the geometrical aspect of the problem, are critical for a correct physical interpretation of the fluid/structure coupling. We prove that the linearization has unique solution.
Energy minimization in two-level dissipative quantum control: Th e integrable case
Bernard Bonnard, Jean-Baptiste Caillau and Olivier Cots
2011, 2011(Special): 198-208 doi: 10.3934/proc.2011.2011.198 +[Abstract](237) +[PDF](2810.0KB)
Abstract:
The aim of this contribution is to re ne some of the computations of [6]. The Lindblad equation modelling a two-level dissipative quantum system is investigated. The control can be interpretated as the action of a laser to rotate a molecule in gas phase, or as the e ect of a magnetic eld on a spin 1=2 particle. For the energy cost, normal extremals of the maximum principle are solution to a three-dimensional Hamiltonian with parameters. The analysis is focussed on an integrable submodel which de nes outside singularities a pseudo-Riemannian metric in dimension ve. Complete quadratures are given for this subcase by means of Weierstra elliptic functions. Preliminary computations of cut and conjugate loci are also provided for a two-dimensional restriction using [9].
Non ordered lower and upper solutions to fourth order problems with functional boundary conditions
Alberto Cabada, João Fialho and Feliz Minhós
2011, 2011(Special): 209-218 doi: 10.3934/proc.2011.2011.209 +[Abstract](223) +[PDF](351.6KB)
Abstract:
In this paper, given $f : I \times (C(I))^2 \times \mathbb{R}^2 \leftarrow \mathbb{R}$ a $L^1$ Carathéodory function, it is considered the functional fourth order equation $u^(iv) (x) = f(x, u, u', u'' (x), u''' (x))$ together with the nonlinear functional boundary conditions $L_0(u, u', u'', u (a)) = 0 = L_1(u, u', u'', u' (a))$ $L_2(u, u', u'', u'' (a), u''' (a)) = 0 = L_3(u, u', u'', u'' (b}, u''' (b)):$ Here $L_i, i$ = 0; 1; 2; 3, are continuous functions satisfying some adequate monotonicity assumptions. It will be proved an existence and location result in presence of non ordered lower and upper solutions and without monotone assumptions on the right hand side of the equation.
Symmetry breaking in problems involving semilinear equations
Lucio Cadeddu and Giovanni Porru
2011, 2011(Special): 219-228 doi: 10.3934/proc.2011.2011.219 +[Abstract](324) +[PDF](346.7KB)
Abstract:
This paper is concerned with two maximization problems where symmetry breaking arises. The rst one consists in the maximization of the energy integral relative to a homogeneous Dirichlet problem governed by the elliptic equation -$\deltau = XF^u^q$ in the annulus $B_(a,a+2)$ of the plane. Here 0  q < 1 and F is a varying subset of $B_(a,a+2)$, with a fixed measure. We prove that a subset which maximizes the corresponding energy integral is not symmetric whenever a is large enough. The second problem we consider is governed by the same equation in a disc $B_a+2$ when $F$ varies in the annulus Ba;a+2 keeping a xed measure. So, now we have a so called maximization problem with a constraint. As in the previous case, we prove that a subset which maximizes the corresponding energy integral is not symmetric whenever a is large enough.
Discrete and differential homotopy in circular restricted three-body control
Jean-Baptiste Caillau, Bilel Daoud and Joseph Gergaud
2011, 2011(Special): 229-239 doi: 10.3934/proc.2011.2011.229 +[Abstract](378) +[PDF](2281.4KB)
Abstract:
The planar circular restricted three-body problem is considered. The control enters linearly in the equation of motion to model the thrust of the third body. The minimum time optimal control problem has two scalar parameters: The ratio of the primaries masses which embeds the two-body problem into the three-body one, and the upper bound on the control norm. Regular extremals of the maximum principle are computed by shooting thanks to continuations with respect to both parameters. Discrete and di erential homotopy are compared in connection with second order sucient conditions in optimal control. Homotopy with respect to control bound gives evidence of various topological structures of extremals.
Decay rate at infinity of the positive solutions of a generalized class of $T$homas-Fermi equations
Santiago Cano-Casanova
2011, 2011(Special): 240-249 doi: 10.3934/proc.2011.2011.240 +[Abstract](237) +[PDF](376.7KB)
Abstract:
This paper ascertains the exact decay rate to zero at in nity of the unique positive solution of the generalized Thomas-Fermi BVP
The problem of global identifiability for systems with tridiagonal matrices
B. Cantó, C. Coll and E. Sánchez
2011, 2011(Special): 250-257 doi: 10.3934/proc.2011.2011.250 +[Abstract](298) +[PDF](263.8KB)
Abstract:
In this work a parametric system with symmetric tridiagonal matrix structure is considered. In particular, parametric systems whose state coecient matrix has non-zero (positive) entries only on the diagonal, the super-diagonal and the sub-diagonal are analyzed. The structural properties of the model are studied, and some conditions to assure the global identi ability are given. These results guarantee the existence of only one solution for the parameters of the system. In practice, systems with this structure arise, for example, via discretization or nite di erence methods for solving boundary and initial value problems involving di erential or partial di erential equations.
A Priori estimates for solutions of differential inclusions
Ovidiu Carja and Victor Postolache
2011, 2011(Special): 258-264 doi: 10.3934/proc.2011.2011.258 +[Abstract](256) +[PDF](315.4KB)
Abstract:
We get a priori estimates for solutions of di erential inclusion $y' \in Ay + F(y)$, where $A$ generates a $C_0$-semigroup in a Banach space and $F$ is a multi-function. The existence of a global solution is also considered.
Finite extinction time property for a delayed linear problem on a manifold without boundary
Alfonso C. Casal, Jesús Ildefonso Díaz and José M. Vegas
2011, 2011(Special): 265-271 doi: 10.3934/proc.2011.2011.265 +[Abstract](292) +[PDF](316.1KB)
Abstract:
We prove that the mere presence of a delayed term is able to connect the initial state u0 on a manifold without boundary
Two nontrivial solutions of a class of elliptic equations with singular term
J. Chen, K. Murillo and E. M. Rocha
2011, 2011(Special): 272-281 doi: 10.3934/proc.2011.2011.272 +[Abstract](242) +[PDF](430.4KB)
Abstract:
We consider the existence of nontrivial solutions of the equation
Cylindrical bending of cusped Reisner-Mindlin plates
Natalia Chinchaladze
2011, 2011(Special): 282-291 doi: 10.3934/proc.2011.2011.282 +[Abstract](233) +[PDF](166.0KB)
Abstract:
Cylindrical bending of cusped Reisner-Mindlin plates are studied. Admissible boundary value problems are investigated. The setting of boundary conditions at the plate edges depends on the geometry of sharpenings of plate edges.
Control of synchrony by delay coupling in complex networks
Chol-Ung Choe, Thomas Dahms, Philipp Hövel and Eckehard Schöll
2011, 2011(Special): 292-301 doi: 10.3934/proc.2011.2011.292 +[Abstract](284) +[PDF](1014.8KB)
Abstract:
Using a master stability function approach, we study synchronization in delay-coupled oscillator networks. The oscillators are modeled by a complex normal form of super- or subcritical Hopf bifurcation. We derive analytical stability conditions and demonstrate that by tuning the phase of the complex coupling constant one can easily control the stability of synchronous periodic states. The phase is identified as a crucial control parameter to switch between in-phase synchronization or desynchronization for general network topologies. For unidirectionally coupled rings or more general networks described by circulant matrices the coupling phase controls in-phase, cluster, or splay states. Our results are robust even for slightly nonidentical oscillators.
Some implications of a new approach to exponential functions on time scales
Jan L. Cieśliński
2011, 2011(Special): 302-311 doi: 10.3934/proc.2011.2011.302 +[Abstract](282) +[PDF](335.4KB)
Abstract:
We present a new approach to exponential functions on time scales and to timescale analogues of ordinary di erential equations. We describe in detail the Cayley-exponential function and associated trigonometric and hyperbolic functions. We show that the Cayley-exponential is related to implicit midpoint and trapezoidal rules, similarly as delta and nabla exponential functions are related to Euler numerical schemes. Extending these results on any Padé approximants, we obtain Pade-exponential functions. Moreover, the exact exponential function on time scales is de fined. Finally, we present applications of the Cayley-exponential function in the $q$-calculus and suggest a general approach to dynamic systems on Lie groups.
Invariant feedback design for control systems with lie symmetries - A kinematic car example
Carsten Collon, Joachim Rudolph and Frank Woittennek
2011, 2011(Special): 312-321 doi: 10.3934/proc.2011.2011.312 +[Abstract](319) +[PDF](206.5KB)
Abstract:
Invariant tracking control design for control systems in state representation with classical Lie point symmetry is considered. The relevance of the invariance aspect is motivated by an exemplary control design for the kinematic car. Introducing the perpendicular distance between the position of the rear axle midpoint and the reference trajectory in combination with the contouring error as tracking error, an invariant feedback w.r.t. the action of $SE$(2) is derived. Generally, the use of compatible tracking errors allows the application of well-known feedback design approaches yielding an error dynamics that is invariant w.r.t. the group action. A normalization procedure allowing the construction of invariant tracking errors is recalled for which a geometric interpretation is presented.
A Bohl-Perron type theorem for random dynamical systems
N. D. Cong, T. S. Doan and S. Siegmund
2011, 2011(Special): 322-331 doi: 10.3934/proc.2011.2011.322 +[Abstract](372) +[PDF](308.0KB)
Abstract:
In this paper, a Bohl-Perron type theorem for random dynamical systems is proved which characterizes uniform exponential stability by means of admissibility. As an application, we obtain a lower bound on the stability radius of random dynamical systems based on the norm of the input-output operator.
Coincidence of Lyapunov exponents and central exponents of linear Ito stochastic differential equations with nondegenerate stochastic term
Nguyen Dinh Cong and Nguyen Thi Thuy Quynh
2011, 2011(Special): 332-342 doi: 10.3934/proc.2011.2011.332 +[Abstract](385) +[PDF](337.4KB)
Abstract:
In this paper we show that under a nondegeneracy condition Lyapunov exponents and central exponents of linear Ito stochastic di erential equation coincide. Furthermore, as the stochastic term is small and tends to zero the highest Lyapunov exponent tends to the highest central exponent of the ordinary di erential equation which is the deterministic part of the system.
Novel formulation of inverse scattering and characterization of scattering data
Francesco Demontis and Cornelis Van der Mee
2011, 2011(Special): 343-350 doi: 10.3934/proc.2011.2011.343 +[Abstract](324) +[PDF](305.1KB)
Abstract:
In this article we formulate the direct and inverse scattering theory for the focusing matrix Zakharov-Shabat system as the construction of a 1, 1-correspondence between focusing potentials with entries in $L^1(\mathbb{R})$ and Marchenko integral kernels, given the fact that these kernels encode the usual scattering data (one reflection coecient, the discrete eigenvalues with positive imaginary part, and the corresponding norming constants) faithfully. In the re ectionless case, we solve the Marchenko equations explicitly using matrix triplets and obtain focusing matrix NLS solutions in closed form.
A linearized system describing stationary incompressible viscous flow around rotating and translating bodies: Improved decay estimates of the velocity and its gradient
Paul Deuring, Stanislav Kračmar and Šárka Nečasová
2011, 2011(Special): 351-361 doi: 10.3934/proc.2011.2011.351 +[Abstract](291) +[PDF](380.7KB)
Abstract:
We consider a linearization of a model for stationary incompressible viscous ow past a rigid body performing a rotation and a translation. Using a representation formula, we obtain pointwise decay bounds for the velocity and its gradient. This result improves estimates obtained by the authors in a previous article.
Numerical computation of normal form coefficients of bifurcations of odes in MATLAB
Virginie De Witte and Willy Govaerts
2011, 2011(Special): 362-372 doi: 10.3934/proc.2011.2011.362 +[Abstract](315) +[PDF](193.6KB)
Abstract:
Normal form coefficients of codim-1 and codim-2 bifurcations of equilibria of ODEs are important since their sign and size determine the bifurcation scenario near the bifurcation points. Multilinear forms with derivatives up to the fifth order are needed in these coefficients. So far, in the Matlab bifurcation software MatCont for ODEs, these derivatives are computed either by finite differences or by symbolic differentiation. However, both approaches have disadvantages. Finite differences do not usually have the required accuracy and for symbolic differentiation the Matlab Symbolic Toolbox is needed. Automatic differentiation is an alternative since this technique is as accurate as symbolic derivatives and no extra software is needed. In this paper, we discuss the pros and cons of these three kinds of differentiation in a specific context by the use of several examples.
About maximum principles for one of the components of solution vector and stability for systems of linear delay differential equations
A. Domoshnitsky
2011, 2011(Special): 373-380 doi: 10.3934/proc.2011.2011.373 +[Abstract](313) +[PDF](265.4KB)
Abstract:
In this paper we de ne and discuss maximum principles for systems of delay di erential equations. Assertions on di erential inequalities for one of the components of solution vector are proven. On this basis new exponential stability results are obtained. In contrast with other results of this sort we do not assume the dominance of the main diagonal in order to get the exponential stability. Various tests of the exponential stability are proposed.
Dispersion relations for supersonic multiple virtual jets
Joshua Du and Liancheng Wang
2011, 2011(Special): 381-390 doi: 10.3934/proc.2011.2011.381 +[Abstract](321) +[PDF](162.2KB)
Abstract:
In this paper, a special type of circulant matrices, circulant trinity matrices, are first introduced and some properties of these matrices are investigated. Then, in studying a supersonic circular N-jet vortex sheet model, circulant trinity matrices are used to classify the general solutions of the $N$-jet model into 2$N$ families. Finally, the dispersion relations of $N$-jets for each family are derived by using Graf’s Addition Theorem.
Bifurcation and bursting in Morris-Lecar model for class I and class II excitability
Lixia Duan, Dehong Zhai and Qishao Lu
2011, 2011(Special): 391-399 doi: 10.3934/proc.2011.2011.391 +[Abstract](382) +[PDF](933.5KB)
Abstract:
In this paper, we study and classify the firing patterns in the Morris-Lecar neuronal model with current-feedback (MLF) control. The Morris-Lecar model has two different types of neuronal excitability (i.e. class I and class II excitability) when the parameters are set appropriately. It is shown that the MLF model exhibits two types of bursting oscillations under the parameter set for class I while exhibits five types of bursting oscillations under the parameter set for class II. Furthermore, we study the relationship between the excitability and bursting oscillations by the two-parameter bifurcation analysis of the fast subsystem for class I and class II excitability, respectively. It shows that different bifurcation structures of the fast subsystem may lead to various types of bursting oscillations in the neuronal model.
Global Cauchy problem of an ideal density-dependent MHD-$\alpha$ model
Jishan Fan and Tohru Ozawa
2011, 2011(Special): 400-409 doi: 10.3934/proc.2011.2011.400 +[Abstract](219) +[PDF](315.7KB)
Abstract:
The global Cauchy problem for an approximation model for the ideal density-dependent MHD- $\alpha$ model is studied. The vanishing limit on is also discussed.
Differential equations of ellipsoidal state estimates in nonlinear control problems under uncertainty
Tatiana Filippova
2011, 2011(Special): 410-419 doi: 10.3934/proc.2011.2011.410 +[Abstract](325) +[PDF](2193.5KB)
Abstract:
The properties of set-valued states of differential control systems with uncertainties in initial data are considered. It is assumed that the dynamical system has a special structure, in which the nonlinear terms in the right-hand sides of related differential equations are quadratic in state coordinates. The model of uncertainty considered here is deterministic, with setmembership description of uncertain items which are taken to be unknown but bounded with given bounds. We construct external and internal ellipsoidal estimates of reachable sets of nonlinear control system and find differential equations of proposed ellipsoidal estimates of reachable sets of nonlinear control system. Numerical simulation results are also given.
Ellipticity of quantum mechanical Hamiltonians in the edge algebra
Heinz-Jürgen Flad and Gohar Harutyunyan
2011, 2011(Special): 420-429 doi: 10.3934/proc.2011.2011.420 +[Abstract](239) +[PDF](377.2KB)
Abstract:
We have studied the ellipticity of quantum mechanical Hamiltonians, in particular of the helium atom, in order to prove existence of a parametrix and corresponding Green operator. The parametrix is considered in local neighbourhoods of coalescence points of two particles. We introduce appropriate hyperspherical coordinates where the singularities of the Coulomb potential are considered as embedded edge/corner-type singularities. This shows that the Hamiltonian can be written as an edge/corner degenerate di erential operator in a pseudo-di erential operator algebra. In the edge degenerate case, we prove the ellipticity of the Hamiltonian.We have studied the ellipticity of quantum mechanical Hamiltonians, in particular of the helium atom, in order to prove existence of a parametrix and corresponding Green operator. The parametrix is considered in local neighbourhoods of coalescence points of two particles. We introduce appropriate hyperspherical coordinates where the singularities of the Coulomb potential are considered as embedded edge/corner-type singularities. This shows that the Hamiltonian can be written as an edge/corner degenerate di erential operator in a pseudo-di erential operator algebra. In the edge degenerate case, we prove the ellipticity of the Hamiltonian.
Global recurrences of multi-time scaled systems
Jean-Pierre Francoise and Claude Piquet
2011, 2011(Special): 430-436 doi: 10.3934/proc.2011.2011.430 +[Abstract](259) +[PDF](357.2KB)
Abstract:
We develop here the rst steps of a long-term investigation on multi-time scaled systems with some spatial heterogeneity. They can be characterized by a slow-variation of \actions" interspaced by local fast variations of \angles". The principle of construction of these systems is presented on a rst example where the actions are periodic solutions of a planar Hamiltonian system. Once the fast perturbation of the angles is added, the whole system displays kind of bursting oscillations (characterized by the alternate of quiescent phases interspaced by fast oscillations) although it is completely integrable. In this rst example, the full analysis of the underlying recurrence is possible. We then discuss a second example which has been discovered by Rossler in the 70s and which inspired this study. This example looks paradigmatic of the complex global recurrences these systems may have.
Variational inequality for the Stokes equations with constraint
Takeshi Fukao
2011, 2011(Special): 437-446 doi: 10.3934/proc.2011.2011.437 +[Abstract](325) +[PDF](354.3KB)
Abstract:
In this paper, the existence problem of the variational inequality for the constrained Stokes equations is considered, in 2- and 3-dimensions with bounded domain. The evolution equation is governed by the subdi erential which formulates the pointwise constraint by the time-dependent obstacle functions. Thanks to the penalty method due to Temam, the hindrance of divergence freeness is avoided. We then characterize the Yosida approximation of the subdi erential and obtain suitable limit conditions.
The geometry of limit cycle bifurcations in polynomial dynamical systems
Valery A. Gaiko
2011, 2011(Special): 447-456 doi: 10.3934/proc.2011.2011.447 +[Abstract](377) +[PDF](322.5KB)
Abstract:
In this paper, applying a canonical system with eld rotation parameters and using geometric properties of the spirals lling the interior and exterior domains of limit cycles, we solve the problem on the maximum number of limit cycles for the classical Lienard polynomial system which is related to the solution of Smale's thirteenth problem. By means of the same geometric approach, we generalize the obtained results and solve the problem on the maximum number of limit cycles surrounding a unique singular point for an arbitrary polynomial system which is related to the solution of Hilbert's sixteenth problem on the maximum number and relative position of limit cycles for planar polynomial dynamical systems.
Phase-field study of solute trapping effect in rapid solidification
P.K. Galenko, E.V. Abramova and D.M. Herlach
2011, 2011(Special): 457-466 doi: 10.3934/proc.2011.2011.457 +[Abstract](259) +[PDF](674.3KB)
Abstract:
The phase-field model of Echebarria, Folch, Karma, and Plapp [Phys. Rev. E 70 (2004) 061604] is extended to the case of rapid solidification in which local non-equilibrium phenomena occur in the bulk phases and within the diffuse solid-liquid interface. Such an extension leads to the fully hyperbolic system of equations given by the atomic diffusion equation and the phase-field equation of motion. This model is applied to the problem of solute trapping, which is accompanied by the entrapment of solute atoms beyond chemical equilibrium by a rapidly moving interface. The model predicts the beginning of complete solute trapping and diffusionless solidification at a finite solidification velocity.
Do longer delays matter? The effect of prolonging delay in CTL activation
Angela Gallegos and Ami Radunskaya
2011, 2011(Special): 467-474 doi: 10.3934/proc.2011.2011.467 +[Abstract](246) +[PDF](500.8KB)
Abstract:
The activation of a speci c immune response takes place in the lymphoid organs such as the spleen. We present here a simpli ed model of the proliferation of speci c immune cells in the form of a single delay equation. We show that the system can undergo switches in stability as the delay is increased, and we interpret these results in the context of sustaining an e ective immune response to a dendritic cell vaccine.
Comparing the efficiency of numerical techniques for the integration of variational equations
Enrico Gerlach and Charlampos Skokos
2011, 2011(Special): 475-484 doi: 10.3934/proc.2011.2011.475 +[Abstract](286) +[PDF](4013.6KB)
Abstract:
We present a comparison of different numerical techniques for the integration of variational equations. The methods presented can be applied to any autonomous Hamiltonian system whose kinetic energy is quadratic in the generalized momenta, and whose potential is a function of the generalized positions. We apply the various techniques to the well-known H´enon-Heiles system, and use the Smaller Alignment Index (SALI) method of chaos detection to evaluate the percentage of its chaotic orbits. The accuracy and the speed of the integration schemes in evaluating this percentage are used to investigate the numerical efficiency of the various techniques.
Transport and generation of macroscopically modulated waves in diatomic chains
Johannes Giannoulis
2011, 2011(Special): 485-494 doi: 10.3934/proc.2011.2011.485 +[Abstract](227) +[PDF](336.6KB)
Abstract:
We derive and justify analytically the dynamics of a small macroscopically modulated amplitude of a single plane wave in a nonlinear diatomic chain with stabilizing on-site potentials including the case where a wave generates another wave via self-interaction. More precisely, we show that in typical chains acoustical waves can generate optical but not acoustical waves, while optical waves are always closed with respect to self-interaction.
$L^1$ maximal regularity for the laplacian and applications
Yoshikazu Giga and Jürgen Saal
2011, 2011(Special): 495-504 doi: 10.3934/proc.2011.2011.495 +[Abstract](300) +[PDF](370.0KB)
Abstract:
Inter alia we prove $L^1$ maximal regularity for the Laplacian in the space of Fourier transformed nite Radon measures FM. This is remarkable, since FM is not a UMD space and by the fact that we obtain $L_p$ maximal regularity for $p$ = 1, which is not even true for the Laplacian in $L^2$. We apply our result in order to construct strong solutions to the Navier-Stokes equations for initial data in FM in a rotating frame. In particular, the obtained results are uniform in the angular velocity of rotation.
Improvement of image processing by using homogeneous neural networks with fractional derivatives theorem
Zbigniew Gomolka, Boguslaw Twarog and Jacek Bartman
2011, 2011(Special): 505-514 doi: 10.3934/proc.2011.2011.505 +[Abstract](234) +[PDF](341.5KB)
Abstract:
The present paper deals with the unique circumvention of designing feed forward neural networks in the task of the interferometry image recog- nition. In order to bring the interferometry techniques to the fore, we recall briefly that this is one of the modern techniques of restitution of three di- mensional shapes of the observed object on the basis of two dimensional flat like images registered by CCD camera. The preliminary stage of this process is conducted with ridges detection, and to solve this computational task the discussed neural network was applied. By looking for the similarities in the biological neural systems authors show the designing process of the homogeneous neural network in the task of maximums detection. The fractional derivative theorem has been involved to assume the weight distribution function as well as transfer functions. To ensure reader that the theoretical considerations are correct, the comprehensive review of experiment results with obtained two dimensional signals have been presented too.
Uniqueness and parameter dependence of positive solutions of third order boundary value problems with $p$-laplacian
John R. Graef and Lingju Kong
2011, 2011(Special): 515-522 doi: 10.3934/proc.2011.2011.515 +[Abstract](302) +[PDF](322.5KB)
Abstract:
We consider the boundary value problem with nonhomogeneous three-point boundary condition Sufficient conditions are obtained for the existence and uniqueness of a positive solution. The dependence of the solution on the parameter $\lambda$ is also studied. This work extends and improves some recent results in the literature on the above problem, especially those in the paper [L. Kong, D. Piao, and L. Wang, Positive solutions for third order boundary value problems with $p$-Laplacian, Result. Math. 55 (2009) 111{128].
Minimization of the number of periodic points for smooth self-maps of closed simply-connected 4-manifolds
Grzegorz Graff and Jerzy Jezierski
2011, 2011(Special): 523-532 doi: 10.3934/proc.2011.2011.523 +[Abstract](213) +[PDF](344.5KB)
Abstract:
Let $M$ be a smooth closed simply-connected $m$-dimensional manifold, $f$ be a smooth self-map of $M$ and $r$ be a given natural number. The invariant $D^m_r [f]$ de fined by the authors in [Forum Math. 21 (2009)] is equal to the minimum of #Fix($g^r$) over all maps $g$ smoothly homotopic to $f$. In this paper we calculate the invariant $D^4_r [f]$ for the class of smooth self-maps of 4-manifolds with fast grow of Lefschetz numbers and for $r$ being a product of di erent primes.
Cauchy problem for a class of nondiagonalizable hyperbolic systems
Todor Gramchev and Nicola Orrú
2011, 2011(Special): 533-542 doi: 10.3934/proc.2011.2011.533 +[Abstract](302) +[PDF](350.0KB)
Abstract:
We investigate the well-posedness of Cauchy problem for weakly hyperbolic systems in one space dimension with time dependent coecients in Sobolev spaces and in the $C^\infty$ category allowing nondiagonalizable principal parts and complex entries in the nilpotent part. We prove well-posedness results by means of an iterative approach under conditions linking the characteristic roots, the entries of the nilpotent part and of the zero order part.
A long-time stable fully discrete approximation of the Cahn-Hilliard equation with inertial term
Maurizio Grasselli, Nicolas Lecoq and Morgan Pierre
2011, 2011(Special): 543-552 doi: 10.3934/proc.2011.2011.543 +[Abstract](291) +[PDF](311.4KB)
Abstract:
We prove the Lyapunov stability of a time and space discretization of the Cahn-Hilliard equation with inertial term. The space discretization is a mixed (or "splitting") nite element method with numerical integration which includes a standard nite di erence approximation. The time discretization is the backward Euler scheme. The smallness assumption on the time step does not depend on the mesh step.
A mathematical model for the spread of streptococcus pneumoniae with transmission due to sequence type
David Greenhalgh, Karen E. Lamb and Chris Robertson
2011, 2011(Special): 553-567 doi: 10.3934/proc.2011.2011.553 +[Abstract](272) +[PDF](274.2KB)
Abstract:
This paper discusses a simple mathematical model to describe the spread of Streptococcus pneumoniae. We suppose that the transmission of the bacterium is determined by multi-locus sequence type. The model includes vaccination and is designed to examine what happens in a vaccinated population if MLSTs can exist as both vaccine and non vaccine serotypes with capsular switching possible from the former to the latter.
   We start off with a discussion of Streptococcus pneumoniae and a review of previous work. We propose a simple mathematical model with two sequence types and then perform an equilibrium and (global) stability analysis on the model. We show that in general there are only three equilibria, the carriage-free equilibrium and two carriage equilibria. If the effective reproduction number $R_e$ is less than or equal to one, then the carriage will die out. If $R_e$ > 1, then the carriage will tend to the carriage equilibrium corresponding to the multi-locus sequence type with the largest transmission parameter. In the case where both multi-locus sequence types have the same transmission parameter then there is a line of carriage equilibria. Provided that carriage is initially present then as time progresses the carriage will approach a point on this line. The results generalize to many competing sequence types. Simulations with realistic parameter values confirm the analytical results.
A natural differential operator on conic spaces
Daniel Grieser
2011, 2011(Special): 568-577 doi: 10.3934/proc.2011.2011.568 +[Abstract](257) +[PDF](327.2KB)
Abstract:
We introduce the notion of a conic space, as a natural structure on a manifold with boundary, and de ne a natural fi rst order di fferential operator, $c_d_\partial$, acting on boundary values of conic one-forms. Conic structures arise, for example, from resolutions of manifolds with conic singularities, embedded in a smooth ambient space. We show that pull-backs of smooth ambient one-forms to the resolution are cd@-closed, and that this is the only local condition on oneforms that is invariantly de ned on conic spaces. The operator $c_d_\partial$ extends to conic Riemannian metrics, and $c_d_\partial$-closed conic metrics have important geometric properties like the existence of an exponential map at the boundary.
Optimal control of pollution stock
Ellina Grigorieva and Evgenii Khailov
2011, 2011(Special): 578-588 doi: 10.3934/proc.2011.2011.578 +[Abstract](264) +[PDF](390.7KB)
Abstract:
A model of an interaction between a manufacturer and the state where the manufacturer produces a single product and the state controls the level of pollution is created and investigated. The model is described by a nonlinear system of two differential equations with two bounded controls. The best optimal strategy is found analytically with the use of the Pontryagin Maximum Principle and Green’s Theorem.
Distributed mathematical models of undetermined "without preference" motion of traffic flow
Sharif E. Guseynov and Shirmail G. Bagirov
2011, 2011(Special): 589-600 doi: 10.3934/proc.2011.2011.589 +[Abstract](231) +[PDF](297.1KB)
Abstract:
Present work proceeds non-deterministic motion of the two-dimensional (2D) vehicular traffic flow, where the traffic flow is assumed as flow of particles in the investigated environment with allowed motion in both forward and opposite directions. Besides, it is assumed that at any fixed time interval in the 2D flow, vehicles could change its positions on the road to any arbitrary placements at the de ned probabilities, even they might be not the neighbouring ones. Such a non-deterministic motion of 2D traffic flow will be named as motion "without preference". Under the pointed assumptions, first it is constructed the non-deterministic discrete mathimatical model, and later by means f using the principle of continuous system there are applied limiting transitions to the constructed discrete model. As a result nondeterministic continuous model in the form of initial-boundary value problem for the integro-di fferential equation is elaborated. In addition probabilistic interpretations of the constructed models and the received results are given.
On continuous models of current stock of divisible productions
Sharif E. Guseynov, Eugene A. Kopytov and Edvin Puzinkevich
2011, 2011(Special): 601-613 doi: 10.3934/proc.2011.2011.601 +[Abstract](310) +[PDF](303.8KB)
Abstract:
In the given paper we investigate the problem of constructing continuous and unsteady mathematical models, to determine the volumes of current stock of divisible productions, in one or several interconnected warehouses using the apparatus of mathematical physics and continuum principle. It is assumed that production distribution and replenishment is continuous. The constructed models are stochastic, and have di erent levels of complexity, adequacy and application potential. The simple model is constructed using the theory of ODE, for construction of more complex models the theory of PDE is applied. Also using additional conditions for the finite-diff erenced model for determination of random volume of divisible homogeneous production is constructed, and this nite di erenced mathematical model makes it possible to determine one of the possible trajectories of the random quantity. All constructed models can be used for on-line monitoring of the dynamics of the random productions volumes.
New regularizing approach to determining the influence coefficient matrix for gas-turbine engines
Sharif E. Guseynov and Sergey M. Yunusov
2011, 2011(Special): 614-623 doi: 10.3934/proc.2011.2011.614 +[Abstract](203) +[PDF](309.2KB)
Abstract:
This paper presents the new approach to the formation of the gas turbine engine diagnostic matrix employing Tikhonov regularization method and taking into account the compressor properties shift under the condition of engine air-gas channel alteration. This method allows eliminating the certain inadequacy of the diagnostic matrices in some cases and removes the restrictions on their implementation for gas turbine engines diagnostics. The elabo- rated regularization algorithm of the calculation-identi cation matrix reversion permits to determine the diagnostic matrix persistently. The suggested method of registration of the compressor properties shift allows providing the adequacy of the engine mathematical model taking into consideration the depreciation of the engine and air-gas channel and consequently obtaining the adequate diagnostic matrix. It is o ered to employ the obtained diagnostic model in the on-board systems of the gas turbine engine control and diagnostics.
3D-2D asymptotic observation for minimization problems associated with degenerate energy-coefficients
Rejeb Hadiji and Ken Shirakawa
2011, 2011(Special): 624-633 doi: 10.3934/proc.2011.2011.624 +[Abstract](208) +[PDF](355.1KB)
Abstract:
In this paper, a class of minimization problems, labeled by an index 0 < $h$ < 1, is considered. Each minimization problem is for a free-energy, motivated by the magnetics in 3D-ferromagnetic thin film, and in the context, the index $h$ denotes the thickness of the observing film. The Main Theorem consists of two themes, which are concerned with the study of the solvability (existence of minimizers) and the 3D-2D asymptotic analysis for our minimization problems. These themes will be discussed under degenerate setting of the material coecients, and such degenerate situation makes the energy-domain be variable with respect to $h$. In conclusion, assuming some restrictive conditions for the domain-variation, a de nite association between our 3D-minimization problems, for very thin $h$, and a 2D-limiting problem, as $h \searrow$ 0, will be demonstrated with help from the theory of $\Gamma$-convergence.
Anisotropic Gevrey regularity for mKdV on the circle
Heather Hannah, A. Alexandrou Himonas and Gerson Petronilho
2011, 2011(Special): 634-642 doi: 10.3934/proc.2011.2011.634 +[Abstract](290) +[PDF](348.7KB)
Abstract:
It is shown that the solution to the Cauchy problem for the modifi ed Korteweg-de Vries equation with initial data in an analytic Gevrey space $G^\sigma$, $\sigma \>= 1$, as a function of the spacial variable belongs to the same Gevrey space. However, considered as function of time the solution does not belong to $G^\sigma$. In fact, it belong to $G^(3\sigma)$ and not to any Gevrey space $G^r$, 1 $\<= r$ < 3$\sigma$.
Pohozaev-Ôtani type inequalities for weak solutions of quasilinear elliptic equations with homogeneous coefficients
Takahiro Hashimoto
2011, 2011(Special): 643-652 doi: 10.3934/proc.2011.2011.643 +[Abstract](247) +[PDF](352.3KB)
Abstract:
In this paper, we are concerned with the following quasilinear el- liptic equations:
Critical second order operators on time scales
Petr Hasil and Petr Zemánek
2011, 2011(Special): 653-659 doi: 10.3934/proc.2011.2011.653 +[Abstract](288) +[PDF](289.1KB)
Abstract:
In this paper we introduce the concept of critical operators for dynamic operators of second order. Next, we show that an arbitrarily small (in a certain sense) negative perturbation of a non-negative critical operator leads to an operator which is no longer non-negative.
Global existence for a functional reaction-diffusion problem from climate modeling
Georg Hetzer
2011, 2011(Special): 660-671 doi: 10.3934/proc.2011.2011.660 +[Abstract](302) +[PDF](376.0KB)
Abstract:
Motivated by coupling an energy balance climate model and a two-species competition model for the bio-sphere, one is led to study the existence of non-negative mild solutions for set-valued functional reaction-diffusion equations involving a memory term and a nonlocal Volterra-type operator. A global existence and boundedness result is established in an m-accretive setting.
Barriers on projective convex sets
Roland Hildebrand
2011, 2011(Special): 672-683 doi: 10.3934/proc.2011.2011.672 +[Abstract](324) +[PDF](325.4KB)
Abstract:
Modern interior-point methods used for optimization on convex sets in ane space are based on the notion of a barrier function. Projective space lacks crucial properties inherent to ane space, and the concept of a barrier function cannot be directly carried over. We present a self-contained theory of barriers on convex sets in projective space which is build upon the projective cross-ratio. Such a projective barrier equips the set with a Codazzi structure, which is a generalization of the Hessian structure induced by a barrier in the ane case. The results provide a new interpretation of the ane theory and serve as a base for constructing a theory of interior-point methods for projective convex optimization.
On general Sturmian theory for abnormal linear Hamiltonian systems
Roman Šimon Hilscher
2011, 2011(Special): 684-691 doi: 10.3934/proc.2011.2011.684 +[Abstract](341) +[PDF](273.7KB)
Abstract:
In this paper we discuss oscillation theory for linear Hamiltonian systems for which we do not impose the controllability (or equivalently normality) assumption. Based on the Sturmian separation and comparison theorems on a compact interval, derived earlier by the author for these systems, we classify them as oscillatory or nonoscillatory. Moreover, we provide comparison theorems for such oscillatory and nonoscillatory systems. One of the goals of this paper is to provide several examples illustrating this new theory.
On an application of fixed point theorem to nonlinear inclusions
Shui-Hung Hou
2011, 2011(Special): 692-697 doi: 10.3934/proc.2011.2011.692 +[Abstract](399) +[PDF](286.4KB)
Abstract:
Sucient conditions for existence of solutions for a class of nonlinear inclusions in in nite dimensional Banach spaces are established. The results are obtained by means of fixed-point theorem for set-valued maps.
On critical exponent for an elliptic equation with non-Lipschitz nonlinearity
Yavdat Il'yasov
2011, 2011(Special): 698-706 doi: 10.3934/proc.2011.2011.698 +[Abstract](264) +[PDF](533.0KB)
Abstract:
This note deals with weak compact support solutions of an elliptic equation with autonomous non-Lipschitz nonlinearity. Using Pohozaev's identity and the spectral analysis with respect to the ber procedure a critical exponent corresponding to the problem is introduced.
Some space-time integrability estimates of the solution for heat equations in two dimensions
Norisuke Ioku
2011, 2011(Special): 707-716 doi: 10.3934/proc.2011.2011.707 +[Abstract](250) +[PDF](360.7KB)
Abstract:
We study some space-time integrability estimates for a solution of an inhomogeneous heat equation in $(0,T) \times \Omega$ with 0-Dirichlet boundary condition, where $\Omega$ is a bounded domain in $\mathbb{R}^2$. We also discuss an exponential integrability estimate for the Poisson equation in $\Omega$ with 0-Dirichlet boundary condition.
Motion of polygonal curved fronts by crystalline motion: v-shaped solutions and eventual monotonicity
Tetsuya Ishiwata
2011, 2011(Special): 717-726 doi: 10.3934/proc.2011.2011.717 +[Abstract](225) +[PDF](158.9KB)
Abstract:
This paper deals with behavior of polygonal plane curves with two asymptotic lines by generalized crystalline curvature flow with a driving force. We show global existence of V-shaped solutions and investigate the deformation patterns of non-V-shaped solutions. We also show that non-V-shaped solutions become V-shaped in finite time.
Global asymptotic stability in a class of nonlinear differential delay equations
Anatoli F. Ivanov and Musa A. Mammadov
2011, 2011(Special): 727-736 doi: 10.3934/proc.2011.2011.727 +[Abstract](360) +[PDF](329.3KB)
Abstract:
An essentially nonlinear di erential equation with delay serving as a mathematical model of several applied problems is considered. Sufficient conditions for the global asymptotic stability of a unique equilibrium are derived. An application to a physiological model by M.C. Mackey is treated in detail.
Dynamic boundary conditions as limit of singularly perturbed parabolic problems
Ángela Jiménez-Casas and Aníbal Rodríguez-Bernal
2011, 2011(Special): 737-746 doi: 10.3934/proc.2011.2011.737 +[Abstract](280) +[PDF](305.0KB)
Abstract:
We obtain dynamic boundary conditions as a limit of parabolic problems with null flux where the time derivative concentrates near the boundary.
Power-law approximation of Bean's critical-state model with displacement current
Frank Jochmann
2011, 2011(Special): 747-753 doi: 10.3934/proc.2011.2011.747 +[Abstract](274) +[PDF](300.9KB)
Abstract:
The paper deals with Bean's critical state model for the description of the electromagnetic field in type-II superconductors in the case where the displacement current is not neglected at least in the surrounding insulating medium. The main goal is the approximation of the Voltage-current relation in the critical state model by a power law.
Asymptotic behavior of Maxwell's equation in one-space dimension with thermal effect
Yuri Kalinin, Volker Reitmann and Nayil Yumaguzin
2011, 2011(Special): 754-762 doi: 10.3934/proc.2011.2011.754 +[Abstract](313) +[PDF](734.8KB)
Abstract:
A coupled system derived from Maxwell’s equations and the heat transfer equation is considered. For this system with perturbations a cocycle formulation is presented. Using Lyapunov functionals the global stability of the zero solution for the autonomous case is shown. In the case of almost periodic perturbations conditions for the existence of almost periodic solutions are derived.
Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions
Barbara Kaltenbacher and Irena Lasiecka
2011, 2011(Special): 763-773 doi: 10.3934/proc.2011.2011.763 +[Abstract](388) +[PDF](386.8KB)
Abstract:
In this paper we show wellposedness of two equations of nonlinear acoustics, as relevant e.g. in applications of high intensity ultrasound. After having studied the Dirichlet problem in previous papers, we here consider Neumann boundary conditions which are of particular practical interest in applications. The Westervelt and the Kuznetsov equation are quasilinear evolutionary wave equations with potential degeneration and strong damping. We prove local in time well-posedness as well as global existence and exponential decay for a slightly modi ed model. A key step of the proof is an appropriate extension of the Neumann boundary data to the interior along with exploitation of singular estimates associated with the analytic semigroup generated by the strongly damped wave equation.
The existence of time global solutions for tumor invasion models with constraints
Risei Kano and Akio Ito
2011, 2011(Special): 774-783 doi: 10.3934/proc.2011.2011.774 +[Abstract](259) +[PDF](305.9KB)
Abstract:
No abstract available.
Singularities in the geometric theory of differential equations
Ulrike Kant and Werner M. Seiler
2011, 2011(Special): 784-793 doi: 10.3934/proc.2011.2011.784 +[Abstract](315) +[PDF](1089.6KB)
Abstract:
We briefly recall the basic ideas of the Vessiot theory, a geometric approach to di fferential equations based on vector fields. Then we show that it allows to extend naturally some results on singularities for ordinary diff erential equations to maximally overdetermined partial diff erential equations.
Reduction of three-dimensional model of the far turbulent wake to one-dimensional problem
Oleg V. Kaptsov and Alexey V. Schmidt
2011, 2011(Special): 794-802 doi: 10.3934/proc.2011.2011.794 +[Abstract](359) +[PDF](444.2KB)
Abstract:
Semi-empirical three-dimensional model of turbulence in the approximation of the far turbulent wake behind a self-propelled body in a passively strati ed medium is considered. The sought quantities are the kinetic turbulent energy, kinetic energy dissipation rate, averaged density defect and density uctuation variance. The full group of transformations admitted by this model is found. The governing equations are reduced into ordinary differential equations by similarity reduction and method of the B-determining equations (BDEs). This system of ordinary di erential equations satisfying natural boundary conditions was solved numerically. The obtained solutions agree with experimental data.
Multi-scale model of bladder cancer development
Eugene Kashdan and Svetlana Bunimovich-Mendrazitsky
2011, 2011(Special): 803-812 doi: 10.3934/proc.2011.2011.803 +[Abstract](259) +[PDF](533.5KB)
Abstract:
Bladder Cancer (BC) is the seventh most common cancer worldwide. Etiology of BC is well known. According to existing statistics, 80% of BC patients had occupational exposure to chemical carcinogens (rubber, dye, textile, or plant industry) or/and were smoking regularly during long periods of time. The carcinogens from the bladder lumen affect umbrella cells of the urothelium (epithelial tissue surrounding bladder) and then subsequently pen- etrate to the deeper layers of the tissue (intermediate and basal cells). It is a years-long process until the carcinogenic substance will accumulate in the tissue in the quantity necessary to trigger DNA mutations leading to the tumor development. We address carcinogen penetration (modeled as a nonlinear diffusion equation with variable coefficient and source term) within the cellular automata (CA) framework of the urothelial cell living cycle. Our approach combines both discrete and continuous models of some of the crucial biological and physical processes inside the urothelium and yields a first theoretical insight on the initial stages of the BC development and growth.
FLUID STRUCTURE INTERACTION PROBLEM WITH CHANGING THICKNESS NON-LINEAR BEAM Fluid structure interaction problem with changing thickness non-linear beam
Emine Kaya, Eugenio Aulisa, Akif Ibragimov and Padmanabhan Seshaiyer
2011, 2011(Special): 813-823 doi: 10.3934/proc.2011.2011.813 +[Abstract](417) +[PDF](391.2KB)
Abstract:
In this work we consider the dynamical response of a non-linear beam interacting with potential flow. The beam part is modeled using a non-linear system of momentum equations for the axial and transverse displacements. Changing in the beam thickness has been modeled in the momentum of inertia. The fluid flow part is subjected to the Bernoulli potential law. In particular we show that for a class of boundary conditions and for a speci c constraint in the beam thickness rate of change, there exists an appropriate energy norm which is bounded by the incoming flow velocity in the liquid region.
Global attractor of the multivalued semigroup associated with a phase-field model of grain boundary motion with constraint
Nobuyuki Kenmochi and Noriaki Yamazaki
2011, 2011(Special): 824-833 doi: 10.3934/proc.2011.2011.824 +[Abstract](299) +[PDF](328.2KB)
Abstract:
We consider a phase-fi eld model of grain boundary motion with constraint, which is a nonlinear system of Kobayashi-Warren-Carter type: a nonlinear parabolic partial diff erential equation and a nonlinear parabolic variational inequality. Recently the existence of solutions to our system was shown in the N-dimensional case. Also the uniqueness was proved in the case when the space dimensional is one and initial data are good. In this paper we study the asymptotic stability of our model without uniqueness. In fact we shall construct global attractors for multivalued semigroups (multivalued semiflows) associated with our system in the N-dimensional case.
A bifurcation result for two point boundary value problem with a strong singularity
Chan-Gyun Kim and Yong-Hoon Lee
2011, 2011(Special): 834-843 doi: 10.3934/proc.2011.2011.834 +[Abstract](343) +[PDF](410.0KB)
Abstract:
We study a global bifurcation phenomena for a degenerate type p-Laplacian problem.
The born approximation and Calderón's method for reconstruction of conductivities in 3-D
Kim Knudsen and Jennifer L. Mueller
2011, 2011(Special): 844-853 doi: 10.3934/proc.2011.2011.844 +[Abstract](396) +[PDF](379.7KB)
Abstract:
Two algorithms for the direct reconstruction of conductivities in a bounded domain in $\mathbb{R}^3$ from surface measurements of the solutions to the conductivity equation are presented. The algorithms are based on complex geometrical optics solutions and a nonlinear scattering transform. We test the algorithms on three numerically simulated examples, including an example with a complex coefficient. The spatial resolution and amplitude of the examples are well-reconstructed.
A stochastic approach for computing the domain of attraction without trajectory simulation
Péter Koltai
2011, 2011(Special): 854-863 doi: 10.3934/proc.2011.2011.854 +[Abstract](243) +[PDF](442.9KB)
Abstract:
We present an approach for the numerical computation of the domain of attraction of some asymptotically stable set for continuous-time autonomous systems. It is based on a set-oriented approximation of the original dynamical system by a Markov jump process. The domain of attraction is extracted from absorption probabilities of the jump process. The method does not perform any trajectory simulation, integrals of the underlying vector eld on the boundary of partition elements are computed instead.
On polyhedral estimates for trajectory tubes of dynamical discrete-time systems with multiplicative uncertainty
Elena K. Kostousova
2011, 2011(Special): 864-873 doi: 10.3934/proc.2011.2011.864 +[Abstract](282) +[PDF](7294.7KB)
Abstract:
The approach for constructing external and internal polyhedral (parallelepiped-valued) estimates of reachable sets and trajectory tubes for the discrete-time systems with a multiplicative uncertainty (for linear systems with the uncertainty in initial states, additive controls and system matrices) is presented. The techniques for set-valued operations using parallelepipeds and parallelotopes as basic sets are described. The solution to an auxiliary problem of finding an internal estimate for the set which is obtained by multiplying an interval matrix on a parallelotope is presented. Recurrence relations for evolution of estimates of reachable sets (cross-sections of trajectory tubes) are described. All proposed estimates can be calculated by explicit formulas. The results of numerical simulations are presented.
Asymptotic of gaps at small coupling and applications of the skew-shift Schrödinger operator
Helge Krüger
2011, 2011(Special): 874-880 doi: 10.3934/proc.2011.2011.874 +[Abstract](280) +[PDF](282.7KB)
Abstract:
I derive an asymptotic formula for the gaps of periodic discrete Schrödinger operators. An application to the skew-shift Schrödinger operator is discussed.
Nonlinear evolution equations associated with mathematical models
Akisato Kubo
2011, 2011(Special): 881-890 doi: 10.3934/proc.2011.2011.881 +[Abstract](211) +[PDF](327.6KB)
Abstract:
We investigate the global existence in time and the asymptotic pro le of solutions of nonlinear evolution equations with strong dissipation. Applying our result to some models of mathematical biology and medicine, we discuss mathematical properties of them.
Periodic solutions for non-isothermal phase transition models
Kota Kumazaki
2011, 2011(Special): 891-902 doi: 10.3934/proc.2011.2011.891 +[Abstract](223) +[PDF](310.2KB)
Abstract:
We study non-isothermal phase transition models of Penrose-Fife type and prove the existence of a periodic solution thereof.
Minimality in iterative systems of Möbius transformations
Petr Kůrka
2011, 2011(Special): 903-912 doi: 10.3934/proc.2011.2011.903 +[Abstract](250) +[PDF](1076.1KB)
Abstract:
We study the parameter space of an iterative system consisting of two hyperbolic disc Möbius transformations. We identify several classes of parameters which yield discrete groups whose fundamental polygons have sides at the Euclidean boundary. It follows that these system are not minimal.
Influence of the distribution component of the magnetic induction vector on rupturing capacity of vacuum switches
Bogdan Kwiatkowski, Tadeusz Kwater and Anna Koziorowska
2011, 2011(Special): 913-921 doi: 10.3934/proc.2011.2011.913 +[Abstract](271) +[PDF](458.0KB)
Abstract:
The paper presents experimental tests results of magnetic field distribution for highcurrent switch. Provided tests show influence of switch shape on field distribution to obtain sustained vitality of vacuum switch. The method shown in the paper let obtain all components of magnetic induction vector. There is aspired to gain the highest value of axial component designing switches shape [1]. It warrants favorable influence on fixity and turn off ability of switch system. There are lower values of peripheral and radial components making bipolar field in space between switches. (The Measurement of magnetic field in highcurrent unipolar switches).
Positive solutions for p-Laplacian equations with concave terms
Sophia Th. Kyritsi and Nikolaos S. Papageorgiou
2011, 2011(Special): 922-930 doi: 10.3934/proc.2011.2011.922 +[Abstract](388) +[PDF](340.7KB)
Abstract:
We consider a nonlinear Dirichlet problem driven by the p-Laplacian diff erential operator, with a nonlinearity concave near the origin and a nonlinear perturbation of it. We look for multiple positive solutions. We consider two distinct cases. One when the perturbation is p-linear and resonant with respect to $\lambda_1 > 0$ (the principal eigenvalue of (-$\Delta_p,W_0^(1,p)(Z)$)) at infi nity and the other when the perturbation is p-superlinear at infi nity. In both cases we obtain two positive smooth solutions. The approach is variational, coupled with the method of upper-lower solutions and with suitable truncation techniques.
Convergence versus periodicity in a single-loop positive-feedback system 1. Convergence to equilibrium
Monica Lazzo and Paul G. Schmidt
2011, 2011(Special): 931-940 doi: 10.3934/proc.2011.2011.931 +[Abstract](244) +[PDF](338.2KB)
Abstract:
We study a parameter-dependent single-loop positive-feedback system in the nonnegative orthant of $\mathbb{R}^n$, with $n\in\mathbb{N}$, that arises in the analysis of the blow-up behavior of large radial solutions of polyharmonic PDEs with power nonlinearities. We describe the global dynamics of the system for arbitrary $n$ and prove that, in every dimension $n\<=4$, all forward-bounded solutions converge to one of two equilibria (one stable, the other unstable). In Part 2 of the paper, we will establish the existence of nontrivial periodic orbits in every dimension $n \>= 12$.
Convergence versus periodicity in a single-loop positive-feedback system 2. Periodic solutions
Monica Lazzo and Paul G. Schmidt
2011, 2011(Special): 941-952 doi: 10.3934/proc.2011.2011.941 +[Abstract](292) +[PDF](358.0KB)
Abstract:
We study a parameter-dependent single-loop positive-feedback system in the nonnegative orthant of $\mathbb{R}^n$, with $n\in\mathbb{N}$, that arises in the analysis of the blow-up behavior of large radial solutions of polyharmonic PDEs with power nonlinearities. In Part 1 of the paper we showed that, in every dimension $n\<=4$, all forward-bounded solutions converge to one of two equilibria (one stable, the other unstable). Here we establish the existence of nontrivial periodic orbits in every dimension $n\>=12$. For $12\<=n\<=16$, we prove that such orbits arise via Hopf bifurcation from the unstable equilibrium. Additional results suggest that at least one Hopf bifurcation occurs whenever $n\>=12$, followed by at least a second one if $n\>=62$, and that the number of successive bifurcations increases without bound as $n\to\infty$.
Numerical approximation of the Chan-Hillard equation with memory effects in the dynamics of phase separation
Nicolas Lecoq, Helena Zapolsky and P.K. Galenko
2011, 2011(Special): 953-962 doi: 10.3934/proc.2011.2011.953 +[Abstract](317) +[PDF](329.2KB)
Abstract:
We consider the modifi ed Cahn-Hilliard equation for phase separation suggested to account for spinodal decomposition in deeply supercooled binary alloy systems or glasses. This equation contains, as additional term, the second-order time derivative of the concentration multiplied by a positive coefficient $\Tau_d$ (time for relaxation). We consider a numerical approximation scheme based on Fourier spectral method and perform numerical analysis of the scheme. We present results of numerical simulations for three spatial dimensions, and examine the stability and convergence of the scheme.
Continuous maximal regularity and analytic semigroups
Jeremy LeCrone and Gieri Simonett
2011, 2011(Special): 963-970 doi: 10.3934/proc.2011.2011.963 +[Abstract](291) +[PDF](327.9KB)
Abstract:
In this paper we establish a result regarding the connection between continuous maximal regularity and generation of analytic semigroups on a pair of densely embedded Banach spaces. More precisely, we show that continuous maximal regularity for a closed operator $A$ : $E_1 \to E_0$ implies that $A$ generates a strongly continuous analytic semigroup on $E_0$ with domain equal $E_1$.
Dynamics of tumor-immune interaction under treatment as an optimal control problem
Urszula Ledzewicz, Mohammad Naghnaeian and Heinz Schättler
2011, 2011(Special): 971-980 doi: 10.3934/proc.2011.2011.971 +[Abstract](260) +[PDF](304.2KB)
Abstract:
For a mathematical model for cancer-immune system interactions under chemotherapy, we consider the problem of moving an initial condition that lies in a region of malignant cancer growth through therapy into a region where cancer regresses and thus control the cancer. We formulate this treatment goal as an optimal control problem and discuss the solutions for various related objective functions in the model.
On optimal singular controls for a general SIR-model with vaccination and treatment
Urszula Ledzewicz and Heinz Schättler
2011, 2011(Special): 981-990 doi: 10.3934/proc.2011.2011.981 +[Abstract](302) +[PDF](302.6KB)
Abstract:
A general SIR-model with vaccination and treatment is considered as a multi-input optimal control problem over a xed time horizon. Existence and local optimality of singular controls is investigated. It is shown that the optimal vaccination schedule can be singular, but that treatment schedules are not.
Spectral analysis for linear differential-algebraic equations
Vu Hoang Linh and Volker Mehrmann
2011, 2011(Special): 991-1000 doi: 10.3934/proc.2011.2011.991 +[Abstract](298) +[PDF](329.8KB)
Abstract:
In this paper, the spectral theory of linear di fferential-algebraic equations (DAEs) is discussed. Lyapunov and Sacker-Sell spectra, which are well known for ordinary diff erential equations (ODEs), are studied for linear DAEs and their adjoint equations. The spectral properties of the DAEs are investigated via the so-called essentially underlying ODEs.
Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness
Shitao Liu and Roberto Triggiani
2011, 2011(Special): 1001-1014 doi: 10.3934/proc.2011.2011.1001 +[Abstract](330) +[PDF](381.6KB)
Abstract:
We study a uniqueness inverse problem for two coupled hyperbolic PDEs with Neumann boundary conditions by means of an additional measurement of Dirichlet boundary traces of the two solutions on a suitable, explicit sub-portion $\Gamma_1$ of the boundary $\Gamma$, and over a computable time interval $T > 0$. Under sharp conditions on $\Gamma_0 = \Gamma\\Gamma_1, T >0$, we establish uniqueness of both the damping and potential coecients for each equation. The proof uses critically the Carleman estimate in [11], together with a suggestion in [8, Thm 8.2.2, p.231]. A Riemannian version would also hold, this time by using the corresponding Carleman estimates in [19].
The existence of weak solutions for a general class of mixed boundary value problems
M.J. Lopez-Herrero
2011, 2011(Special): 1015-1024 doi: 10.3934/proc.2011.2011.1015 +[Abstract](315) +[PDF](165.8KB)
Abstract:
This paper shows the existence of weak solutions for a generalized class of boundary value problems with indefinite potentials which remain outside the general scope of the classical theorems of L. G a rding [3] and P. D. Lax and A. N. Milgram [6].
Bounds for blow-up time in nonlinear parabolic systems
Monica Marras and Stella Vernier Piro
2011, 2011(Special): 1025-1031 doi: 10.3934/proc.2011.2011.1025 +[Abstract](260) +[PDF](305.9KB)
Abstract:
We consider a nonlinear parabolic system with Neumann boundary conditions which solution may blow up in nite time $t*$ We determine a lower bound for $t*$ by using a Sobolev type inequality. In addition an upper bound for $t*$ is obtained, under alternative conditions on the non linearities.
Firing map of an almost periodic input function
Wacław Marzantowicz and Justyna Signerska
2011, 2011(Special): 1032-1041 doi: 10.3934/proc.2011.2011.1032 +[Abstract](420) +[PDF](370.1KB)
Abstract:
In mathematical biology and the theory of electric networks the fi ring map of an integrate-and-fi re system is a notion of importance. In order to prove useful properties of this map authors of previous papers assumed that the stimulus function $f$ of the system $ẋ$ = $f(t, x)$ is continuous and usually periodic in the time variable. In this work we show that the required properties of the firing map for the simplifi ed model $ẋ$ = $f(t)$ still hold if $f \in L(^1_(loc))(R)$ and $f$ is an almost periodic function. Moreover, in this way we prepare a formal framework for next study of a discrete dynamics of the firing map arising from almost periodic stimulus that gives information on consecutive resets (spikes).
Positivity, robust stability and comparison of dynamic systems
Alexey G. Mazko
2011, 2011(Special): 1042-1051 doi: 10.3934/proc.2011.2011.1042 +[Abstract](271) +[PDF](337.3KB)
Abstract:
We study generalized classes of positive and monotone dynamic systems in a partially ordered Banach space. Using results from the nonlinear operators theory, we establish new algebraic conditions for stability of equilibrium states of a class of monotone-type di fferential and diff erence systems. Conditions for the positivity and absolute stability of diff erential systems with delay are proposed. Using new technique for constructing the invariant sets of diff erential systems, we generalize known positivity conditions for linear and nonlinear diff erential systems with respect to typical classes of cones. In addition, we generalize the comparison principle for a finite set of diff erential systems and formulate robust stability conditions for some families of diff erential systems in terms of cone inequalities.
Strong solutions of doubly nonlinear Navier-Stokes equations
Jochen Merker
2011, 2011(Special): 1052-1060 doi: 10.3934/proc.2011.2011.1052 +[Abstract](273) +[PDF](337.3KB)
Abstract:
In this article existence of weak and strong solutions to doubly nonlinear incompressible Navier-Stokes equations

$(\partialb(u))/(\partialt)+$div$(b(u)\times u) = $-$d\pi +$div$\(a(\nabla^(sym)u)) + f,$     div$(u)=0,$
is discussed, where $u$ models the velocity vector field of a homogeneous non-Newtonian fluid whose momentum $b(u)$ depends nonlinearly on $u$.
A necessary and sufficient condition for oscillation of second order sublinear delay dynamic equations
RazIye Mert and A. Zafer
2011, 2011(Special): 1061-1067 doi: 10.3934/proc.2011.2011.1061 +[Abstract](319) +[PDF](299.4KB)
Abstract:
Time scale calculus approach allows one to treat the continuous, discrete, as well as more general systems simultaneously. In this article we use this tool to establish a necessary and sucient condition for the oscillation of a class of second order sublinear delay dynamic equations on time scales. Some well known results in the literature are improved and extended.
Periodic solutions for some fully nonlinear fourth order differential equations
Feliz Minhós
2011, 2011(Special): 1068-1077 doi: 10.3934/proc.2011.2011.1068 +[Abstract](318) +[PDF](333.1KB)
Abstract:
In this paper we present sufficient conditions for the existence of solutions to the periodic fourth order boundary value problem

$u^((4))(x) = f(x,u(x),u'(x),u''(x),u'''(x))$
$u^((i))(a) = u^((i))(b), i=0,1,2,3,$
for $x \in [a,b],$ and $f : [a,b] \times \mathbb{R}^4\to\mathbb{R}$ a continuous function. To the best of our knowledge it is the first time where this type of general nonlinearities is considered in fourth order equations with periodic boundary conditions.
 The difficulties in the odd derivatives are overcome due to the following arguments: the control on the third derivative is done by a Nagumo-type condition and the bounds on the first derivative are obtained by lower and upper solutions, not necessarily ordered.
 By this technique, not only it is proved the existence of a periodic solution, but also, some qualitative properties of the solution can be obtained.
Gradient systems on networks
Delio Mugnolo and René Pröpper
2011, 2011(Special): 1078-1090 doi: 10.3934/proc.2011.2011.1078 +[Abstract](286) +[PDF](423.6KB)
Abstract:
We consider a class of linear di erential operators acting on vector-valued function spaces with general coupled boundary conditions. Unlike in the more usual case of so-called quantum graphs, the boundary conditions can be nonlinear. After introducing a suitable Lyapunov function we prove well-posedness and invariance results for the corresponding nonlinear di usion problem.
Constructing approximate solutions for three dimensional stationary Stokes flow
Frank Müller and Werner Varnhorn
2011, 2011(Special): 1091-1100 doi: 10.3934/proc.2011.2011.1091 +[Abstract](282) +[PDF](306.6KB)
Abstract:
In this article we develop a method for the numerical solution of a boundary value problem for the system of Stokes equations in three dimensions. Simultaneously we develop the method for linear splines and for quasi interpolants. For this we start with a representation of the uniquely determined velocity field $v$ by the double layer potential. In a fi rst step we approximate the kernel an the source density of the double layer potential. In a second step we use a collocation method for the determination of the unknown values of the source density in the collocation points. The convergence analysis is carried out and error estimates are given.
Optimal control problems for quasi-variational inequalities and its numerical approximation
Yusuke Murase, Atsushi Kadoya and Nobuyuki Kenmochi
2011, 2011(Special): 1101-1110 doi: 10.3934/proc.2011.2011.1101 +[Abstract](318) +[PDF](312.2KB)
Abstract:
The main objective of this paper is to discuss about optimal control problems in which the state equations may have multiple solutions. Our state equations are represented by the so-called quasi-variational inequalities and some difficulties for the mathematical treatment arise from such a structure. From the numerical point of view we propose a class of regular approximations for them in which state equations are uniquely solved and the control spaces are relaxed, and further a class of their time-discretizations which are schemes of usual elliptic optimal control problems.
Oscillating solutions to a parabolic-elliptic system related to a chemotaxis model
Yūki Naito and Takasi Senba
2011, 2011(Special): 1111-1118 doi: 10.3934/proc.2011.2011.1111 +[Abstract](260) +[PDF](300.3KB)
Abstract:
In this paper, we consider solutions to a Cauchy problem for a parabolic-elliptic system in two dimensional space. This system is a simpli ed version of a chemotaxis model, and is also a model of self-interacting particles. The behavior of solutions to the problem closely depends on the $L^1$-norm of the solutions. If the quantity is larger than 8$\pi$, the solution blows up in nite time. If the quantity is smaller than the critical mass, the solution exists globally in time. In the critical case, in nite blowup solutions are found. In the present paper, we direct our attention to radial solutions to the problem whose $L^1$-norm is equal to 8$\pi$ and nd oscillating solutions.
On the global stability of an SIRS epidemic model with distributed delays
Yukihiko Nakata, Yoichi Enatsu and Yoshiaki Muroya
2011, 2011(Special): 1119-1128 doi: 10.3934/proc.2011.2011.1119 +[Abstract](340) +[PDF](381.9KB)
Abstract:
In this paper, we establish the global asymptotic stability of an endemic equilibrium for an SIRS epidemic model with distributed time delays. It is shown that the global stability holds for any rate of immunity loss, if the basic reproduction number is greater than 1 and less than or equals to a critical value. Otherwise, there is a maximal rate of immunity loss which guarantees the global stability. By using an extension of a Lyapunov functional established by [C.C. McCluskey, Complete global stability for an SIR epidemic model with delay-Distributed or discrete, Nonlinear Anal. RWA. 11 (2010) 55-59], we provide a partial answer to an open problem whether the endemic equilibrium is globally stable, whenever it exists, or not.
Multiple bounded variation solutions of a capillarity problem
Franco Obersnel and Pierpaolo Omari
2011, 2011(Special): 1129-1137 doi: 10.3934/proc.2011.2011.1129 +[Abstract](293) +[PDF](331.3KB)
Abstract:
We discuss existence and multiplicity of bounded variation solutions of the non-homogeneous Neumann problem for the prescribed mean curvature equation

-div$(\nabla u/\sqrt(1+|\nablau|^2))=g(x,u)+h$     in $\Omega$
-$\nablau*v/\sqrt(1+|\nablau|^2)=k$         on $\partial\Omega$
where $g(x, s)$ is periodic with respect to $s$. Our approach is variational and makes use of non-smooth critical point theory in the space of bounded variation functions.
On a result of C.V. Coffman and W.K. Ziemer about the prescribed mean curvature equation
Franco Obersnel and Pierpaolo Omari
2011, 2011(Special): 1138-1147 doi: 10.3934/proc.2011.2011.1138 +[Abstract](299) +[PDF](332.3KB)
Abstract:
We produce a detailed proof of a result of C.V. Co ffman and W.K. Ziemer [1] on the existence of positive solutions of the Dirichlet problem for the prescribed mean curvature equation

-div$(\nablau/\sqrt(1+|\nablau|^2)=\lambdaf(x,u)$ in $\Omega,$     $u=0$ on $\partial\Omega$
assuming that $f$ has a superlinear behaviour at $u = 0$.
Two-parameter locus of boundary crisis: Mind the gaps!
Hinke M. Osinga and James Rankin
2011, 2011(Special): 1148-1157 doi: 10.3934/proc.2011.2011.1148 +[Abstract](267) +[PDF](561.6KB)
Abstract:
Boundary crisis is a mechanism for destroying a chaotic attractor when one parameter is varied. In a two-parameter setting the locus of boundary crisis is associated with curves of homo- or heteroclinic tangency bifurcations of saddle periodic orbits. It is known that the locus of boundary crisis contains many gaps, corresponding to channels (regions of positive measure) where a non-chaotic attractor persists. One side of such a subduction channel is a saddle-node bifurcation of a periodic orbit that marks the start of a periodic window in the chaotic regime; the other side of the channel is formed by a homo- or heteroclinic tangency bifurcation associated with the saddle periodic orbit involved in the saddle-node bifurcation. We present a two-parameter study of boundary crisis in the Ikeda map, which models the dynamics of energy levels in a laser ring cavity. We confirm the existence of many gaps on the boundary- crisis locus. However, the gaps correspond to subduction channels that can have a rather different structure compared to what is known in the literature.
Determination of motion from orbit in the three-body problem
Hiroshi Ozaki, Hiroshi Fukuda and Toshiaki Fujiwara
2011, 2011(Special): 1158-1166 doi: 10.3934/proc.2011.2011.1158 +[Abstract](311) +[PDF](455.0KB)
Abstract:
We discuss the equal mass three-body motion in which the shape of the orbit is given. The conservation of the center of mass and a constant of motion (the total angular momentum or the total energy) leads to the uniqueness of the equal mass three-body motion in given some sorts of orbits. Although the proof was already published on an article by the present authors in 2009, here we give some complementary explanations. We show that, even in the unequal mass three-body periodic motions in which each of bodies draws its own orbit, the shape of the orbits, conservation of the center of mass and a constant of motion provide some candidates of the motion of three bodies. The reality of the motion should be tested whether the equation of motion is satisfied or not. Even if the three bodies draw unclosed orbits, we can show that similarly.
Second order oscillation of mixed nonlinear dynamic equations with several positive and negative coefficients
Abdullah Özbekler and A. Zafer
2011, 2011(Special): 1167-1175 doi: 10.3934/proc.2011.2011.1167 +[Abstract](264) +[PDF](321.3KB)
Abstract:
New oscillation criteria are obtained for superlinear and sublinear forced dynamic equations having positive and negative coecients by means of nonprincipal solutions.
Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion
J. F. Padial
2011, 2011(Special): 1176-1185 doi: 10.3934/proc.2011.2011.1176 +[Abstract](296) +[PDF](317.8KB)
Abstract:
We consider an inverse 2D free-boundary elliptic-parabolic problem, modeling the transient regime of a magnetically con ned plasma in a non ideal Stellarator device. The inverse nature of the problem comes from the fact that the associated Grad-Shafranov equation involves some unknown nonlinear terms which must be determined by the current-carrying Stellarator condition. One of the main diculties for the mathematical approach is due to the fact that these nonlinear terms are neither a monotone function and are nor a Lipschitz function either. Nevertheless, we can introduce an auxiliary problem for which the solution of our original problem is a supersolution. On the other hand, we can to obtain a subsolution for this new problem. Finally, by applying the comparison principle for quasi-linear problems, we obtain some appropriate estimates on the location and size of the plasma region.
A proximal-like algorithm for vibro-impact problems with a non-smooth set of constraints
Laetitia Paoli
2011, 2011(Special): 1186-1195 doi: 10.3934/proc.2011.2011.1186 +[Abstract](252) +[PDF](167.6KB)
Abstract:
We consider a discrete mechanical system subjected to perfect unilateral contraints characterized by some geometrical inequalities $f_\alpha(q) >=0$, $\alpha \in {1,...,v}$, with $v >=1$. We assume that the transmission of the velocities at impacts is governed by a Newton’s impact law with a restitution coefficient $e \in [0, 1]$, allowing for conservation of kinetic energy if $e=1$, or loss of kinetic energy if $e \in [0, 1)$, when the constraints are saturated. Starting from a formulation of the dynamics as a first order measure-differential inclusion for the unknown velocities, time-stepping schemes inspired by the proximal methods can be proposed. Convergence results in the single-constraint case $(v = 1)$ are recalled and extended to the multi-constraint case $(v > 1)$, leading to new existence results for this kind of problems.
Energy-minimal transfers in the vicinity of the lagrangian point $L_1$
Gautier Picot
2011, 2011(Special): 1196-1205 doi: 10.3934/proc.2011.2011.1196 +[Abstract](332) +[PDF](462.1KB)
Abstract:
This article deals with the problem of computing energy-minimal trajectories between the invariant manifolds in the neighborhood of the equilibrium point $L_1$ of the restricted 3-body problem. Initializing a simple shooting method with solutions of the corresponding linear optimal control problem, we numerically compute energy-minimal extremals from the Pontryagin's Maximum principle, whose optimality is ensured thanks to the second order optimality condition.
Parabolic Liouville-type theorems via their elliptic counterparts
Pavol Quittner and Philippe Souplet
2011, 2011(Special): 1206-1213 doi: 10.3934/proc.2011.2011.1206 +[Abstract](264) +[PDF](303.9KB)
Abstract:
We study several parabolic problems with gradient structure and show how the nonexistence of entire stationary solutions can be used in the proof of nonexistence of entire time-dependent solutions.
Physical statistical modelling of bending vibrations
Nils Raabe and Claus Weihs
2011, 2011(Special): 1214-1223 doi: 10.3934/proc.2011.2011.1214 +[Abstract](256) +[PDF](1536.4KB)
Abstract:
One serious problem in deep-hole drilling is the formation of a dynamic disturbance called spiralling which causes holes with several lobes. One explanation for the occurrence of spiralling is the intersection of time varying bending eigenfrequencies with multiples of the rotational frequency of the boring bar leading to a regenerative e ect. This e ect results from the periodical tilt of the drillhead cutting in each lobe after each revolution and continues in a self exciting manner even when the original causing eigenfrequency keeps changing. We propose a physical-statistical model consisting of a system of coupled di erential equations and allowing the explicit Maximum Likelihood estimation of the modal parameters and by this the implicit estimation of the bending eigenfrequency courses. An extensive simulation for the evaluation of the properties of these estimators and tted courses has now been conducted. It is shown that the results of the model can be improved by tting polynomial local regressions frequency band wise. With the tted eigenfrequency courses it is possible to set up the machining parameters in a way that intersections of speci c eigenfrequencies with multiples of the rotational frequency and spiralling correspondingly get unlikely.
Dynamics of dendrite growth in a binary alloy with magnetic field effect
Amer Rasheed, Aziz Belmiloudi and Fabrice Mahé
2011, 2011(Special): 1224-1233 doi: 10.3934/proc.2011.2011.1224 +[Abstract](262) +[PDF](1131.6KB)
Abstract:
In order to analyze the dynamical changes in the free dendritic growth due to magnetic- field, we have developed a 2D phase- field model which consists of nonlinear evolutive and coupled systems of flow, concentration and phase field in an isothermal environment. We present the realistic numerical simulations of the influence of various magnetic- fields and other critical parameters of the derived model on the evolution of dendrites during the solidifi cation of the binary mixture of Nickel-Copper (Ni-Cu).
Global regular solutions to the Navier-Stokes equations with large flux
Joanna Rencławowicz and Wojciech M. Zajączkowski
2011, 2011(Special): 1234-1243 doi: 10.3934/proc.2011.2011.1234 +[Abstract](308) +[PDF](334.1KB)
Abstract:
We consider the Navier-Stokes motion in a bounded cylinder with boundary slip conditions. We assume an inflow and an outflow of the fluid through the bottom and the top of the cylinder where the magnitude of the flux is not restricted. We require that the derivatives of the initial velocity and the external force with respect to the variable along the axis of the cylinder are sufficiently small. Under these conditions we are able to prove global existence of regular solutions. Since we are interested in nonvanishing in time flux we need to use the Hopf function to derive global energy estimate.
Numerical study of secondary heteroclinic bifurcations near non-reversible homoclinic snaking
Thorsten Riess
2011, 2011(Special): 1244-1253 doi: 10.3934/proc.2011.2011.1244 +[Abstract](229) +[PDF](869.8KB)
Abstract:
We discuss the emergence of isolas of secondary heteroclinic bifurcations near a non-reversible homoclinic snaking curve in parameter space that is generated by a codimension-one equilibrium-to-periodic (EtoP) heteroclinic cycle. We use a numerical method based on Lin's method to compute and continue these secondary heteroclinic EtoP orbits for a well-known system.
Classification of a class of relative equilibria in three body coulomb systems
Florian Rupp and Jürgen Scheurle
2011, 2011(Special): 1254-1262 doi: 10.3934/proc.2011.2011.1254 +[Abstract](224) +[PDF](389.6KB)
Abstract:
We consider a class of relative equilibria of a non-regularized ensemble of three charged particles in the fully three-dimensional Euclidean space, i.e., uniform rotational motions of (i) one negative and one positive point charge as well as the “classical” case of (ii) two negative point charges in the electrostatic field of a fixed positive point charge, such that the mutual distances between the three particles and the axis of rotation stay constant over time. Depending on the physical parameters this kind of relative equilibria of such systems are completely classified. Thereby, all our considerations deal with arbitrary positive and negative charge values of the particles as well as arbitrary values for the masses of the freely movable ones.
Dispersive type estimates for fourier integrals and applications to hyperbolic systems
Michael Ruzhansky and Jens Wirth
2011, 2011(Special): 1263-1270 doi: 10.3934/proc.2011.2011.1263 +[Abstract](327) +[PDF](354.7KB)
Abstract:
In this note we announce dispersive estimates for Fourier integrals with parameter-dependent phase functions in terms of geometric quantities of associated families of Fresnel surfaces. The results are based on a multidimensional van der Corput lemma due to the rst author.
    Applications to dispersive estimates for hyperbolic systems and scalar higher order hyperbolic equations are also discussed.
Particular solution to the Euler-Cauchy equation with polynomial non-homegeneities
Adnan H. Sabuwala and Doreen De Leon
2011, 2011(Special): 1271-1278 doi: 10.3934/proc.2011.2011.1271 +[Abstract](257) +[PDF](258.8KB)
Abstract:
The Euler-Cauchy differential equation is one of the first, and simplest, forms of a higher order non-constant coefficient ordinary di erential equation that is encountered in an undergraduate differential equations course. For a non-homogeneous Euler-Cauchy equation, the particular solution is typically determined by either using the method of variation of parameters or transforming the equation to a constant-coefficient equation and applying the method of undetermined coefficients. This paper demonstrates the surprising form of the particular solution for the most general n$^(th)$ order Euler-Cauchy equation when the non-homogeneity is a polynomial. In addition, a formula that can be used to compute the unknown coecients in the form of the particular solution is presented.
Random graph and stochastic process contributions to network dynamics
Deena Schmidt, Janet Best and Mark S. Blumberg
2011, 2011(Special): 1279-1288 doi: 10.3934/proc.2011.2011.1279 +[Abstract](342) +[PDF](253.4KB)
Abstract:
A fundamental question regarding neural systems and other applications involving networks is the extent to which the network architecture may contribute to the observed dynamics. We explore this question by investigating two different processes on three different graph structures, asking to what extent the graph structure (as represented by the degree distribution) is reflected in the dynamics of the process. Our findings suggest that memoryless processes are more likely to reflect the degree distribution, whereas processes with memory can robustly give power law or other heavy-tailed distributions regardless of degree distribution.
Duck farming on the two-torus: Multiple canard cycles in generic slow-fast systems
Ilya Schurov
2011, 2011(Special): 1289-1298 doi: 10.3934/proc.2011.2011.1289 +[Abstract](245) +[PDF](222.1KB)
Abstract:
Generic slow-fast systems with only one (time-scaling) parameter on the two-torus have attracting canard cycles for arbitrary small values of this parameter. This is in drastic contrast with the planar case, where canards usually occur in two-parametric families. In present work, general case of nonconvex slow curve with several fold points is considered. The number of canard cycles in such systems can be effectively computed and is no more than the number of fold points. This estimate is sharp for every system from some explicitly constructed open set.
Stochastic wave equations with cubic nonlinearity and Q-regular additive noise in $\mathbb{R}^2$
Henri Schurz
2011, 2011(Special): 1299-1308 doi: 10.3934/proc.2011.2011.1299 +[Abstract](256) +[PDF](336.6KB)
Abstract:
Semi-linear wave equations on rectangular domains in $\mathbb{R}^2$ (vibrating plates) with certain cubic quasi-nonlinearities and perturbed by a Q-regular space-time white noise are considered analytically. These models as 2nd order SPDEs (stochastic partial differential equations) with non-random Dirichlet- type boundary conditions describe the displacement of noisy vibrations of rectangular plates as met in engineering. We discuss their analysis by the eigen- function approach allowing us to truncate the infinite-dimensional stochastic systems (i.e. the SDEs of Fourier coefficients related to semilinear SPDEs), to control its energy, existence, uniqueness, continuity and stability. A conservation law for at most linearly growing expected energy is established in terms of system-parameters.
Superradiance problem in a 3D annular domain
Indranil SenGupta, Weisheng Jiang, Bo Sun and Maria Christina Mariani
2011, 2011(Special): 1309-1318 doi: 10.3934/proc.2011.2011.1309 +[Abstract](221) +[PDF](370.9KB)
Abstract:
Superradiance is an important phenomena in quantum mechanics which has many practical applications. Recently the superradiance integral equation in three-dimensional balls has been extensively studied. In this paper we consider the superradiance integral equation over an annulus. A differential operator that commutes with the radial part of the superradiance integral equation is found. A complete orthogonal basis for the problem is derived. A generalization is given for the problem.
Existence and continuity of strong solutions of partly dissipative reaction diffusion systems
Zhoude Shao
2011, 2011(Special): 1319-1328 doi: 10.3934/proc.2011.2011.1319 +[Abstract](345) +[PDF](340.1KB)
Abstract:
We discuss the existence and continuity of strong solutions of partly dissipative reaction diffusion systems of the FitzHugh-Nagumo type. Under appropriate conditions, we proved the existence of strong solutions of such systems on $[0, \infty)$ using a Galerkin type of argument. Then we proved that these strong solutions are continuous with respect to initial data in the space $V \times H^1 (\Omega)$, where $V$ is a subspace of $H^1 (\Omega)$ defined according to the boundary condition imposed for the $u$- component in our system. The continuity result is independent of the spatial dimension $n$.
Standing waves in a complex Ginzburg-Landau equation with time-delay feedback
Michael Stich and Carsten Beta
2011, 2011(Special): 1329-1334 doi: 10.3934/proc.2011.2011.1329 +[Abstract](319) +[PDF](114.5KB)
Abstract:
Standing waves are studied as solutions of a complex Ginzburg- Landau equation subjected to local and global time-delay feedback terms. The onset of standing waves is studied at the instability of the homogeneous periodic solution with respect to spatially periodic perturbations. The solution of this spatiotemporal wave pattern is given and is compared to the homogeneous periodic solution.
Regularity criteria in weak spaces in terms of the pressure to the MHD equations
Tomoyuki Suzuki
2011, 2011(Special): 1335-1343 doi: 10.3934/proc.2011.2011.1335 +[Abstract](285) +[PDF](331.6KB)
Abstract:
We study regularity of weak solutions to the viscous incompressible magnetohydrodynamic equations in $\mathbb{R}^3 \times (0, T)$. We give regularity criteria for weak solutions in terms of the pressure and the magnetic elds in Lorentz spaces.
Global bifurcation structure on a shadow system with a source term - Representation of all solutions-
Hideaki Takaichi, Izumi Takagi and Shoji Yotsutani
2011, 2011(Special): 1344-1350 doi: 10.3934/proc.2011.2011.1344 +[Abstract](252) +[PDF](269.8KB)
Abstract:
As the first step to understand the Gierer-Meinhardt system with source term, it is important to know the global bifurcation diagram of a shadow system. For the case without source term, it is well-understood. However, for the case with source term, the shadow system has a nonlocal term. Thus standard methods do not work, and there are a few partial results even for one-dimensional case. We give explicit representations of all solutions in terms of elliptic functions. They play crucial roles to clarify the global bifurcation diagram.
Sharp blow-up for semilinear wave equations with non-compactly supported data
Hiroyuki Takamura, Hiroshi Uesaka and Kyouhei Wakasa
2011, 2011(Special): 1351-1357 doi: 10.3934/proc.2011.2011.1351 +[Abstract](279) +[PDF](318.3KB)
Abstract:
This paper corrects Asakura's observation on semilinear wave equations with non-compactly supported data by showing a sharp blow-up theorem for classical solutions. We know that there is no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak, in spite of nite propagation speed of the linear wave. Our theorem clari es the final criterion on such a phenomenon.
Global existence of solutions for higher order nonlinear damped wave equations
Hiroshi Takeda
2011, 2011(Special): 1358-1367 doi: 10.3934/proc.2011.2011.1358 +[Abstract](250) +[PDF](328.5KB)
Abstract:
We consider a Cauchy problem for a polyharmonic nonlinear damped wave equation. We obtain a critical condition of the nonlinear term to ensure the global existence of solutions for small data. Moreover, we show the op-timal decay property of solutions under the sharp condition on the nonlinear exponents, which is a natural extension of the results for the nonlinear damped wave equations. The proof is based on $L^p-L^q$ type estimates of the fundamental solutions of the linear polyharmonic damped wave equations.
Traveling fronts in perturbed multistable reaction-diffusion equations
Masaharu Taniguchi
2011, 2011(Special): 1368-1377 doi: 10.3934/proc.2011.2011.1368 +[Abstract](264) +[PDF](316.4KB)
Abstract:
In this paper we study the existence and non-existence of traveling front solutions in multistable reaction-di usion equations. If this equation has a traveling front solution, a perturbed equation also has a traveling front solution. We study how the speed and the traveling pro le depend on nonlinear terms.
Is the Trudinger-Moser nonlinearity a true critical nonlinearity?
Kyril Tintarev
2011, 2011(Special): 1378-1384 doi: 10.3934/proc.2011.2011.1378 +[Abstract](303) +[PDF](334.4KB)
Abstract:
While the critical nonlinearity $\int |u|^2^$* for the Sobolev space $H^1$ in dimension N > 2 lacks weak continuity at any point, Trudinger-Moser nonlinearity $\int e^(4\piu^2)$ in dimension N = 2 is weakly continuous at any point except zero. In the former case the lack of weak continuity can be attributed to invariance with respect to actions of translations and dilations. The Sobolev space $H^1_0$f the unit disk $\mathbb{D}\subset\mathbb{R}^2$ possesses transformations analogous to translations (Möbius transformations) and nonlinear dilations $r\to r^s$. We present improvements of the Trudinger-Moser inequality with nonlinearities sharper than $\int e^(4\piu^2)$ that lack weak continuity at any point and possess (separately), translation and dilation invariance. We show, however, that no nonlinearity of the form $\int F(|x|, u(x))dx$ is both dilation- and Möbius shift-invariant. The paper also gives a new, very short proof of the conformal-invariant Trudinger-Moser inequality obtained recently by Mancini and Sandeep [10] and of a sharper version of Onofri-type inequality of Beckner [4].
Polarization dynamics during takeover collisions of solitons in systems of coupled nonlinears Schödinger equations
M. D. Todorov
2011, 2011(Special): 1385-1394 doi: 10.3934/proc.2011.2011.1385 +[Abstract](259) +[PDF](801.5KB)
Abstract:
For the system of coupled nonlinear Schrödinger equations we investigate numerically the takeover interaction dynamics of elliptically polarized solitons. In the case of general elliptic polarization, analytical solution for the shapes of a steadily propagating solitons are not available, and we develop a numerical algorithm finding the shape. We use the superposition of generally elliptical polarized solitons as the initial condition for investigating the soliton dynamics. In order to extract the pure effect of the initial phase angle, we consider the case without cross-modulation – the Manakov system. The sum of the masses for the two quasi-particles is constant and the total pseudomementum and energy of the system are conserved. In the case of nontrivial cross-modulation combining it with different initial phase angles causes velocity shifts of interacted solitons. The results of this work outline the role of the initial phase, initial polarization and the interplay between them and nonlinear couplings on the interaction dynamics of solitons in system of coupled nonlinear Schrödinger equations.
Stability, boundedness and uniform boundedness of solutions of nonlinear delay differential equations
Cemil Tunç
2011, 2011(Special): 1395-1403 doi: 10.3934/proc.2011.2011.1395 +[Abstract](390) +[PDF](295.7KB)
Abstract:
In this paper, we consider a Lienard equation with multiple variable deviating arguments. By using the Lyapunov second (direct) method, we discuss the stability, boundedness and uniform boundedness of solutions of the equation considered. An example is given to illustrate the feasibility of the proposed results.
Wavelet analysis of phase clusters in a distributed biochemical system
Andrey Yu. Verisokin, Darya V. Verveyko, Eugene B. Postnikov and Anastasia I. Lavrova
2011, 2011(Special): 1404-1412 doi: 10.3934/proc.2011.2011.1404 +[Abstract](324) +[PDF](468.5KB)
Abstract:
The spatio-temporal dynamics of glycolysis in distributed medium have been studied both theoretically and experimentally. Different patterns such as travelling waves, standing waves and clusters have been observed in experiment. We describe pattern formation using distributed Selkov model that describes kinetics of phosphofructokinase which is a key enzyme of glycolytic reactions. We have found in numerical simulations that the varying the diffusion coefficient values within the range of 0−10$^$−$^$3 shows a large variety of phase patterns: from the birth of a hierarchy of phase clusters to their complete phase synchronization. In order to understand the mechanism phase clusters emergence and their dynamics we are using continuous wavelet transform.
Generalizations of Naismith's problem: Minimal transit time between two points in a heterogenous terrian
Erik I. Verriest
2011, 2011(Special): 1413-1422 doi: 10.3934/proc.2011.2011.1413 +[Abstract](255) +[PDF](834.3KB)
Abstract:
Naismith obtained a set of empirical rules for the time required to move through a terrain. In this paper we solve the problem of determining the path which minimizes the transit time between two points on a given terrain. We give an interpretation of Naismith’s rule which leads to an elegant geometric construction of the optimal solution. This problem is a paradigm for the navigation of an autonomous vehicle in a heterogenous terrain.
Application of He's method to the modified Rayleigh equation
Darya V. Verveyko and Andrey Yu. Verisokin
2011, 2011(Special): 1423-1431 doi: 10.3934/proc.2011.2011.1423 +[Abstract](246) +[PDF](156.3KB)
Abstract:
In this work we analyze the application of He’s variational method for an estimation of limit cycles and oscillation periods for the class of self-sustained oscillations described by the modified Rayleigh equation. The main goal of the research is to find suitable trial functions which allow to reproduce the period of limit-cycle motion with a high degree of accuracy. There is an especial consideration of the Selkov model in the modified Rayleigh form having only one extremum that does not allow to apply the classical method of slow and fast motions. In this case He’s method allows to find the period of a limit-cycle motion with a high accuracy and to predict its value for various parameters of the concerned equations. Thus, it is possible to assert that at a correct choice of trial function the considered method gives exact results not only in the case of harmonic oscillations but also in the case of relaxation ones.
Harmonic limits of dynamical systems
Tobias Wichtrey
2011, 2011(Special): 1432-1439 doi: 10.3934/proc.2011.2011.1432 +[Abstract](313) +[PDF](324.7KB)
Abstract:
In this paper, we analyze the rotational behaviour of dynamical systems, particulary of solutions of ODEs. With rotational behaviour we mean the existence of rotational factor maps, i. e., semi-conjugations to rotations in the complex plane. In order to analyze this kind of rotational behaviour, we introduce harmonic limits lim$_(T\to\infty)1/T\int^T_0 e^(itw)f(\Phi_tx)$dt. We discuss the connection between harmonic limits and rotational factor maps, and some properties of the limits, e. g., existence under the presence of an invariant measure by the Wiener-Wintner Ergodic Theorem. Finally, we look at linear differential equations (autonomous and periodic), and show the connection between the frequencies of the rotational factor maps and the imaginary parts of the eigenvalues of the system matrix (or of the Floquet exponents in the periodic case).
Bifurcation analysis of complex bursting induced by two different time-scale slow variables
Zhuoqin Yang and Tingting Guan
2011, 2011(Special): 1440-1447 doi: 10.3934/proc.2011.2011.1440 +[Abstract](252) +[PDF](800.8KB)
Abstract:
In Wierschem and Bertram model describing bursting modulated by slow glycolytic oscillation, different complex bursting patterns are produced by interaction of two slow variables with different time scales. Generation mechanisms of the complex bursting patterns with one or multiple short bursts and a long burst, are investigated by an extended fast/slow analysis, when a faster slow variable is considered as a bifurcation parameter of fast subsystem, while a slower slow variable only has an effect on bifurcation curves of the fast subsystem.
Cylindrical blowup solutions to the isothermal Euler-Poisson equations
Manwai Yuen
2011, 2011(Special): 1448-1456 doi: 10.3934/proc.2011.2011.1448 +[Abstract](297) +[PDF](161.4KB)
Abstract:
This proceeding is the continued version of the analytical blowup solutions for 2-dimensional Euler-Poisson equations in ”M.W. Yuen, Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars, J. Math. Anal. Appl. 341 (2008), 445–456”. With the extension of the blowup solutions with radial symmetry for the isothermal Euler-Poisson equations in R$^2$, the cylindrical blowup solutions in R$^N$ ($N>=3$) with are constructed by the separation method. Here, the constructed 3-dimensional blowup solutions could be applied to interpret the evolution of cylindrical cloud for star formation in astrophysics.
Periodic oscillations in a class of fuzzy neural networks under impulsive control
Yong Zhao and Qishao Lu
2011, 2011(Special): 1457-1466 doi: 10.3934/proc.2011.2011.1457 +[Abstract](444) +[PDF](297.0KB)
Abstract:
In this paper, a class of fuzzy neural networks under impulsive control is discussed. Using Mawhin's continuation theorem of coincidence degree and the Halanay-type inequalities, some sucient conditions for existence and globally exponential stability of periodic solution are established. Fuzzy rules are put into Lipschitzian condition and a new method to study stability is introduced, based on the Halanay-type inequalities. Furthermore, some neural network models are considered to illustrate the feasibility and e ectiveness of our results.
Inverse problems for linear ill-posed differential-algebraic equations with uncertain parameters
Sergiy Zhuk
2011, 2011(Special): 1467-1476 doi: 10.3934/proc.2011.2011.1467 +[Abstract](287) +[PDF](229.1KB)
Abstract:
This paper describes a minimax state estimation approach for linear differential-algebraic equations (DAEs) with uncertain parameters. The approach addresses continuous-time DAEs with non-stationary rectangular matrices and uncertain bounded deterministic input. An observation’s noise is supposed to be random with zero mean and unknown bounded correlation function. Main result is a Generalized Kalman Duality (GKD) principle, describing a dual control problem. Main consequence of the GKD is an optimal minimax state estimation algorithm for DAEs with non-stationary rectangular matrices. An algorithm is illustrated by a numerical example for 2D timevarying DAE with a singular matrix pencil.

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