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Volume 1, 2008

Discrete & Continuous Dynamical Systems - S

2008 , Volume 1 , Issue 1

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On sharp interface limits of Allen--Cahn/Cahn--Hilliard variational inequalities
John W. Barrett , Harald Garcke and  Robert Nürnberg
2008, 1(1): 1-14 doi: 10.3934/dcdss.2008.1.1 +[Abstract](52) +[PDF](298.7KB)
Using formally matched asymptotic expansions we identify the sharp interface asymptotic limit of an Allen-Cahn/Cahn-Hilliard system using a novel approach which enables us to handle the case of variational inequalities.
Counting uniformly attracting solutions of nonautonomous differential equations
Arno Berger
2008, 1(1): 15-25 doi: 10.3934/dcdss.2008.1.15 +[Abstract](36) +[PDF](192.9KB)
Bounded uniform attractors and repellors are the natural nonautonomous analogues of autonomous stable and unstable equilibria. Unlike for equilibria, it is generally a difficult dynamical task to determine the number of uniformly attracting or repelling solutions for a given nonautonomous equation, even if the latter exhibits strong structural properties such as e.g. polynomial growth in space or periodicity in time. The present note highlights this aspect by proving that the number of uniform attractors is locally finite for several classes of equations, and by providing examples for which this number can be any $N\in \N$. These results and examples extend and complement recent work on nonautonomous differential equations.
On the κ - θ model of cellular flames: Existence in the large and asymptotics
Claude-Michel Brauner , Michael L. Frankel , Josephus Hulshof , Alessandra Lunardi and  G. Sivashinsky
2008, 1(1): 27-39 doi: 10.3934/dcdss.2008.1.27 +[Abstract](34) +[PDF](570.8KB)
We consider the κ - θ model of flamefront dynamics introduced in [6]. We show that a space-periodic problem for the lattersystem of two equations is globally well-posed. We prove that nearthe instability threshold the front is arbitrarily close to thesolution of the Kuramoto-Sivashinsky equation on a fixed timeinterval if the evolution starts from close configurations.The dynamics generated by the model isillustrated by direct numerical simulation.
Incompressible fluids with shear rate and pressure dependent viscosity: Regularity of steady planar flows
M. Bulíček and  P. Kaplický
2008, 1(1): 41-50 doi: 10.3934/dcdss.2008.1.41 +[Abstract](76) +[PDF](194.3KB)
We study the regularity of steady planar flow of fluids where the shearing stress may depend on the symmetric part of the velocity vector field and the pressure. For simplicity the periodic boundary conditions are considered. Using Meyers estimates we show that there exists a solution which is smooth. In the case where it is allowed to test weak formulation of the problem with a weak solution we prove regularity of all weak solutions.
Control of travelling walls in a ferromagnetic nanowire
Gilles Carbou , Stéphane Labbé and  Emmanuel Trélat
2008, 1(1): 51-59 doi: 10.3934/dcdss.2008.1.51 +[Abstract](48) +[PDF](153.0KB)
We investigate the problem of controlling the magnetic moment in a ferromagnetic nanowire submitted to an external magnetic field in the direction of the nanowire. The system is modeled with the one dimensional Landau-Lifschitz equation. In the absence of control, there exist particular solutions, which happen to be relevant for practical issues, called travelling walls. In this paper, we prove that it is possible to move from a given travelling wall profile to any other one, by acting on the external magnetic field. The control laws are simple and explicit, and the resulting trajectories are shown to be stable.
A mathematical analysis of a dynamical frictional contact model in thermoviscoelasticity
Oanh Chau , R. Oujja and  Mohamed Rochdi
2008, 1(1): 61-70 doi: 10.3934/dcdss.2008.1.61 +[Abstract](41) +[PDF](163.3KB)
We are interested in the dynamic evolution of a thermoviscoelastic body which is on frictional contact with a rigid foundation. The contact is modeled by a general normal damped response condition with friction law and heat exchange. We establish the existence and uniqueness of the weak solution, under the condition that the viscosity is sufficiently strong. Finally the numerical analysis of a fully discrete scheme is presented.
On radial solutions of semi-relativistic Hartree equations
Yonggeun Cho and  Tohru Ozawa
2008, 1(1): 71-82 doi: 10.3934/dcdss.2008.1.71 +[Abstract](43) +[PDF](198.6KB)
We consider the semi-relativistic Hartree type equation with nonlocal nonlinearity $F(u) = \lambda (|x|^{-\gamma} * |u|^2)u, 0 < \gamma < n, n \ge 1$. In [2, 3], the global well-posedness (GWP) was shown for the value of $\gamma \in (0, \frac{2n}{n+1}), n \ge 2$ with large data and $\gamma \in (2, n), n \ge 3$ with small data. In this paper, we extend the previous GWP result to the case for $\gamma \in (1, \frac{2n-1}n), n \ge 2$ with radially symmetric large data. Solutions in a weighted Sobolev space are also studied.
Eventual local positivity for a biharmonic heat equation in RN
Filippo Gazzola and  Hans-Christoph Grunau
2008, 1(1): 83-87 doi: 10.3934/dcdss.2008.1.83 +[Abstract](45) +[PDF](110.9KB)
We study the positivity preserving property for the Cauchy problem for the linear fourth order heat equation. Although the complete positivity preserving property fails, we show that it holds eventually on compact sets.
Positive solutions of a third order nonlocal boundary value problem
John R. Graef and  Bo Yang
2008, 1(1): 89-97 doi: 10.3934/dcdss.2008.1.89 +[Abstract](41) +[PDF](139.2KB)
We consider a nonlocal boundary value problem for a third order differential equation. Sufficient conditions for the existence and nonexistence of positive solutions for the problem are obtained. The results are illustrated with some examples.
Positive solutions of some nonlinear BVPs involving singularities and integral BCs
G. Infante
2008, 1(1): 99-106 doi: 10.3934/dcdss.2008.1.99 +[Abstract](32) +[PDF](148.3KB)
In this paper we discuss the existence of positive solutions of some nonlocal boundary value problems subject to integral boundary conditions and where the involved nonlinearity might be singular.
Arnold tongues for bifurcation from infinity
Victor S. Kozyakin , Alexander M. Krasnosel’skii and  Dmitrii I. Rachinskii
2008, 1(1): 107-116 doi: 10.3934/dcdss.2008.1.107 +[Abstract](44) +[PDF](205.6KB)
We consider discrete time systems $x_{k+1}=U(x_{k};\lambda)$, $x\in\R^{N}$, with a complex parameter $\lambda$. The map $U(\cdot;\lambda)$ at infinity contains a principal linear term, a bounded positively homogeneous nonlinearity, and a smaller part. We describe the sets of parameter values for which the large-amplitude $n$-periodic trajectories exist for a fixed $n$. In the related problems on small periodic orbits near zero, similarly defined parameter sets, known as Arnold tongues, are more narrow.
A class of hemivariational inequalities for electroelastic contact problems with slip dependent friction
Stanislaw Migórski
2008, 1(1): 117-126 doi: 10.3934/dcdss.2008.1.117 +[Abstract](36) +[PDF](185.0KB)
In this paper we deal with a class of inequality problems for static frictional contact between a piezoelastic body and a foundation. The constitutive law is assumed to be electrostatic and involves a nonlinear elasticity operator. The friction condition is described by the Clarke subdifferential relations of nonmonotone and multivalued character in the tangential directions on the boundary. We derive a variational formulation which is a coupled system of a hemivariational inequality and an elliptic equation. The existence of solutions to the model is a consequence of a more general result obtained from the theory of pseudomonotone mappings.
Higher order two-point boundary value problems with asymmetric growth
Feliz Minhós and  A. I. Santos
2008, 1(1): 127-137 doi: 10.3934/dcdss.2008.1.127 +[Abstract](35) +[PDF](183.4KB)
In this work it is studied the higher order nonlinear equation

$\u^{( n)} (x)=f(x,u(x),u^{'}(x),\ldots ,u^{( n-1)} (x)) $

with $n\in \mathbb{N}$ such that $n\geq 2,$ $f:[ a,b] \times \mathbb{R}^{n}\rightarrow \mathbb{R}$ a continuous function, and the two-point boundary conditions

$u^{(i)}(a) =A_{i},\text{ \ \ }A_{i}\in \mathbb{R},\text{ \ }i=0,\ldots ,n-3$,
$u^{( n-1) }(a) =u^{( n-1) }(b)=0.$

From one-sided Nagumo-type condition, allowing that $f$ can be unbounded, it is obtained an existence and location result, that is, besides the existence, given by Leray-Schauder topological degree, some bounds on the solution and its derivatives till order $(n-2)$ are given by well ordered lower and upper solutions.
    An application to a continuous model of human-spine, via beam theory, will be presented.

Global existence for a wave equation on $R^n$
Perikles G. Papadopoulos and  Nikolaos M. Stavrakakis
2008, 1(1): 139-149 doi: 10.3934/dcdss.2008.1.139 +[Abstract](28) +[PDF](177.0KB)
We study the initial value problem for some degenerate non-linear dissipative wave equations of Kirchhoff type: $ u_{t t}-\phi (x)||\grad u(t)||^{2\gamma}\Delta u+\delta u_{t} = f(u),x\in R^n,t\geq 0,$ with initial conditions $ u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $N \geq 3, delta > 0, \gamma\geq 1$, $f(u)=|u|^{a}u$ with $a>0$ and $(\phi (x))^{-1} =g (x)$ is a positive function lying in $L^{N/2}(R^n)\cap L^{\infty}(R^n)$. If the initial data $\{ u_{0},u_{1}\}$ are small and $||\grad u_{0}||>0$, then the unique solution exists globally and has certain decay properties.
3D steady compressible Navier--Stokes equations
Milan Pokorný and  Piotr B. Mucha
2008, 1(1): 151-163 doi: 10.3934/dcdss.2008.1.151 +[Abstract](31) +[PDF](192.3KB)
We study the steady compressible Navier--Stokes equations in a bounded smooth three-dimensional domain, together with the slip boundary conditions. We show that for a certain class of the pressure laws, there exists a weak solution with bounded density (in $L^\infty$ up to boundary).
Two-equation model of mean flow resonances in subcritical flow systems
Sergey A. Suslov
2008, 1(1): 165-176 doi: 10.3934/dcdss.2008.1.165 +[Abstract](34) +[PDF](212.6KB)
Amplitude equations of Landau type, which describe the dynamics of the most amplified periodic disturbance waves in slightly supercritical flow systems, have been known to form reliable and sufficiently accurate low-dimensional models capable of predicting the asymptotic magnitude of saturated perturbations. However the derivation of similar models for estimating the threshold disturbance amplitude in subcritical systems faces multiple resonances which lead to the singularity of model coefficients. The observed resonances are traced back to the interaction between the mean flow distortion induced by the decaying fundamental disturbance harmonic and other decaying disturbance modes. Here we suggest a methodology of deriving a two-equation dynamical system of coupled cubic amplitude equations with non-singular coefficients which resolve the resonances and are capable of predicting the threshold amplitude for weakly nonlinear subcritical regimes. The suggested reduction procedure is based on the consistent use of an appropriate orthogonality condition which is different from a conventional solvability condition. As an example, a developed procedure is applied to a system of Navier-Stokes equations describing a subcritical plane Poiseuille flow. Predictions of the so-developed model are found to be in reasonable agreement with experimentally detected threshold amplitudes reported in literature.
Uniqueness of the principal eigenvalue in nonlocal boundary value problems
J. R. L. Webb
2008, 1(1): 177-186 doi: 10.3934/dcdss.2008.1.177 +[Abstract](159) +[PDF](172.4KB)
In the study of nonlinear boundary value problems, existence of a positive solution can be shown if the nonlinearity 'crosses' the principal eigenvalue, the eigenvalue corresponding to a positive eigenfunction. It is well known that such an eigenvalue is unique for symmetric problems but it was unclear for general nonlocal boundary conditions. Here some old results due to Krasnosel'skiĭ are applied to show that the nonlocal problems which have been well studied over the last few years do have a unique principal eigenvalue. Some estimates and some comparison results are also given.

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