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Discrete & Continuous Dynamical Systems - A

2013 , Volume 33 , Issue 11&12

Special issue
Jerry Goldstein on the occasion of his 70th birthday

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Preface
Gisèle Ruiz Goldstein and Alain Miranville
2013, 33(11&12): i-ii doi: 10.3934/dcds.2013.33.11i +[Abstract](442) +[PDF](83.8KB)
Abstract:
Jerome Arthur Goldstein (Jerry) was born on August 5, 1941 in Pittsburgh, PA. He attended Carnegie Mellon University, then called Carnegie Institute of Technology, where he earned his Bachelors of Science degree (1963), Masters of Science degree (1964) and Ph.D. (1967) in Mathematics. Jerry took a postdoctoral position at the Institute for Advanced Study in Princeton, followed by an Assistant Professorship at Tulane University. He became a Full Professor at Tulane University in 1975. After twenty-four years there, Jerry moved ``upriver" to join the faculty (and his wife, Gisèle) at Louisiana State University, where he was Professor of Mathematics from 1991 to 1996. In 1996 Jerry and Gisèle moved to the University of Memphis.

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Floquet representations and asymptotic behavior of periodic evolution families
Fatih Bayazit, Ulrich Groh and Rainer Nagel
2013, 33(11&12): 4795-4810 doi: 10.3934/dcds.2013.33.4795 +[Abstract](509) +[PDF](375.9KB)
Abstract:
We use semigroup techniques to describe the asymptotic behavior of contractive, periodic evolution families on Hilbert spaces. In particular, we show that such evolution families converge almost weakly to a Floquet representation with discrete spectrum.
Singularity formation and blowup of complex-valued solutions of the modified KdV equation
Jerry L. Bona, Stéphane Vento and Fred B. Weissler
2013, 33(11&12): 4811-4840 doi: 10.3934/dcds.2013.33.4811 +[Abstract](557) +[PDF](515.9KB)
Abstract:
The dynamics of the poles of the two--soliton solutions of the modified Korteweg--de Vries equation $$ u_t + 6u^2u_x + u_{xxx} = 0 $$ are investigated. A consequence of this study is the existence of classes of smooth, complex--valued solutions of this equation, defined for $-\infty < x < \infty$, exponentially decreasing to zero as $|x| \to \infty$, that blow up in finite time.
Periodic traveling--wave solutions of nonlinear dispersive evolution equations
Hongqiu Chen and Jerry L. Bona
2013, 33(11&12): 4841-4873 doi: 10.3934/dcds.2013.33.4841 +[Abstract](665) +[PDF](506.5KB)
Abstract:
For a general class of nonlinear, dispersive wave equations, existence of periodic, traveling-wave solutions is studied. These traveling waveforms are the analog of the classical cnoidal-wave solutions of the Korteweg-de Vries equation. They are determined to be stable to perturbation of the same period. Their large wavelength limit is shown to be solitary waves.
On the asymptotic behavior of variational inequalities set in cylinders
Michel Chipot and Karen Yeressian
2013, 33(11&12): 4875-4890 doi: 10.3934/dcds.2013.33.4875 +[Abstract](527) +[PDF](359.5KB)
Abstract:
We study the asymptotic behavior of solutions to variational inequalities with pointwise constraint on the value and gradient of the functions as the domain becomes unbounded. First, as a model problem, we consider the case when the constraint is only on the value of the functions. Then we consider the more general case of constraint also on the gradient. At the end we consider the case when there is no force term which corresponds to Saint-Venant principle for linear problems.
Hopf bifurcation for a size-structured model with resting phase
Jixun Chu and Pierre Magal
2013, 33(11&12): 4891-4921 doi: 10.3934/dcds.2013.33.4891 +[Abstract](612) +[PDF](1659.1KB)
Abstract:
This article investigates Hopf bifurcation for a size-structured population dynamic model that is designed to describe size dispersion among individuals in a given population. This model has a nonlinear and nonlocal boundary condition. We reformulate the problem as an abstract non-densely defined Cauchy problem, and study it in the frame work of integrated semigroup theory. We prove a Hopf bifurcation theorem and we present some numerical simulations to support our analysis.
On a Dirichlet problem in bounded domains with singular nonlinearity
Giuseppe Maria Coclite and Mario Michele Coclite
2013, 33(11&12): 4923-4944 doi: 10.3934/dcds.2013.33.4923 +[Abstract](515) +[PDF](428.1KB)
Abstract:
In this paper we prove the existence and regularity of positive solutions of the homogeneous Dirichlet problem \begin{equation*} -Δ u=g(x,u)     in     \Omega,         u=0    on     ∂ \Omega, \end{equation*} where $g(x,u)$ can be singular as $u\rightarrow0^+$ and $0\le g(x,u)\le\frac{\varphi_0(x)}{u^p}$ or $0\le$ $ g(x,u)$ $\le$ $\varphi_0(x)(1+\frac{1}{u^p})$, with $\varphi_0 \in L^m(\Omega), 1 ≤ m.$ There are no assumptions on the monotonicity of $g(x,\cdot)$ and the existence of super- or sub-solutions.
Ultraparabolic equations with nonlocal delayed boundary conditions
Gabriella Di Blasio
2013, 33(11&12): 4945-4965 doi: 10.3934/dcds.2013.33.4945 +[Abstract](506) +[PDF](424.5KB)
Abstract:
A class of ultraparabolic equations with delay, arising from age--structured population diffusion, is analyzed. For such equations well--posedness as well as regularity results with respect to the space variables are proved.
Boundary value problem for elliptic differential equations in non-commutative cases
Angelo Favini, Rabah Labbas, Stéphane Maingot and Maëlis Meisner
2013, 33(11&12): 4967-4990 doi: 10.3934/dcds.2013.33.4967 +[Abstract](519) +[PDF](390.8KB)
Abstract:
This paper is devoted to abstract second order complete elliptic differential equations set on $\left[ 0,1\right] $ in non-commutative cases. Existence, uniqueness and maximal regularity of the strict solution are proved. The study is performed in $C^{\theta }\left( \left[ 0,1\right] ;X\right) $.
Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations
Angelo Favini, Rabah Labbas, Keddour Lemrabet, Stéphane Maingot and Hassan D. Sidibé
2013, 33(11&12): 4991-5014 doi: 10.3934/dcds.2013.33.4991 +[Abstract](493) +[PDF](449.6KB)
Abstract:
In this work, a biharmonic equation with an impedance (non standard) boundary condition and more general equations are considered. The study is performed in the space $L^{p}(-1,0$ $;$ $X)$, $1 < p < \infty $, where $X$ is a UMD Banach space. The problem is obtained through a formal limiting process on a family of boundary and transmission problems $(P^{\delta})_{\delta > 0}$ set in a domain having a thin layer. The limiting problem models, for instance, the bending of a thin plate with a stiffness on a part of its boundary (see Favini et al. [13]).
    We build an explicit representation of the solution, then we study its regularity and give a meaning to the non standard boundary condition.
Local Hadamard well--posedness and blow--up for reaction--diffusion equations with non--linear dynamical boundary conditions
Alessio Fiscella and Enzo Vitillaro
2013, 33(11&12): 5015-5047 doi: 10.3934/dcds.2013.33.5015 +[Abstract](428) +[PDF](705.2KB)
Abstract:
The paper deals with local well--posedness, global existence and blow--up results for reaction--diffusion equations coupled with nonlinear dynamical boundary conditions. The typical problem studied is \[\begin{cases} u_{t}-\Delta u=|u|^{p-2} u        in   (0,\infty)\times\Omega,\\ u=0                    on     [0,\infty) \times \Gamma_{0},\\ \frac{\partial u}{\partial\nu} = -|u_{t}|^{m-2}u_{t}       on   [0,\infty)\times\Gamma_{1},\\ u(0,x)=u_{0}(x)         in   \Omega \end{cases}\] where $\Omega$ is a bounded open regular domain of $\mathbb{R}^{n}$ ($n\geq 1$), $\partial\Omega=\Gamma_0\cup\Gamma_1$, $2\le p\le 1+2^*/2$, $m>1$ and $u_0\in H^1(\Omega)$, ${u_0}_{|\Gamma_0}=0$. After showing local well--posedness in the Hadamard sense we give global existence and blow--up results when $\Gamma_0$ has positive surface measure. Moreover we discuss the generalization of the above mentioned results to more general problems where the terms $|u|^{p-2}u$ and $|u_{t}|^{m-2}u_{t}$ are respectively replaced by $f\left(x,u\right)$ and $Q(t,x,u_t)$ under suitable assumptions on them.
On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces
Simona Fornaro and Abdelaziz Rhandi
2013, 33(11&12): 5049-5058 doi: 10.3934/dcds.2013.33.5049 +[Abstract](477) +[PDF](361.8KB)
Abstract:
In this paper we give sufficient conditions ensuring that the space of test functions $C_c^{\infty}(R^N)$ is a core for the operator $$L_0u=\Delta u-Mx\cdot \nabla u+\frac{\alpha}{|x|^2}u=:Lu+\frac{\alpha}{|x|^2}u,$$ and $L_0$ with domain $W_\mu^{2,p}(R^N)$ generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p_\mu(R^N),\,1 < p < \infty$. Here $M$ is a positive definite $N\times N$ hermitian matrix and $\mu$ is the unique invariant measure for the Ornstein-Uhlenbeck operator $L$. The proofs are based on an $L^p$-weighted Hardy's inequality and perturbation techniques.
Prey-predator models with infected prey and predators
J. Gani and R. J. Swift
2013, 33(11&12): 5059-5066 doi: 10.3934/dcds.2013.33.5059 +[Abstract](578) +[PDF](328.7KB)
Abstract:
Some deterministic models for prey and predators are considered, when both may become infected, the infection of the prey being either of the SIS or SIR type. We also study a simplified model for surviving predators.
On a class of model Hilbert spaces
Fritz Gesztesy, Rudi Weikard and Maxim Zinchenko
2013, 33(11&12): 5067-5088 doi: 10.3934/dcds.2013.33.5067 +[Abstract](461) +[PDF](532.3KB)
Abstract:
A detailed description of the model Hilbert space $L^2(\mathbb{R}; d\Sigma; K)$, where $K$ represents a complex, separable Hilbert space, and $\Sigma$ denotes a bounded operator-valued measure, is provided. In particular, we show that several alternative approaches to such a construction in the literature are equivalent.
    These spaces are of fundamental importance in the context of perturbation theory of self-adjoint extensions of symmetric operators, and the spectral theory of ordinary differential operators with operator-valued coefficients.
Nonlocal phase-field systems with general potentials
Maurizio Grasselli and Giulio Schimperna
2013, 33(11&12): 5089-5106 doi: 10.3934/dcds.2013.33.5089 +[Abstract](453) +[PDF](487.3KB)
Abstract:
We consider a phase-field model of Caginalp type where the free energy depends on the order parameter $\chi$ in a nonlocal way. Therefore, the resulting system consists of the energy balance equation coupled with a nonlinear and nonlocal ODE for $\chi$. Such system has been analyzed by several authors, in particular when the configuration potential is a smooth double-well function. More recently, the first author has established the existence of a finite-dimensional global attractor in the case of a potential defined on $(-1,1)$ and singular at the endpoints. Here we examine both the case of regular potentials as well as the case of physically more relevant singular potentials (e.g., logarithmic). We prove well-posedness results and the eventual global boundedness of solutions uniformly with respect to the initial data. In addition, we show that the separation property holds in the case of singular potentials. Thanks to these results, we are able to demonstrate the existence of a finite-dimensional global attractor in the present cases as well.
Partial reconstruction of the source term in a linear parabolic initial problem with Dirichlet boundary conditions
Davide Guidetti
2013, 33(11&12): 5107-5141 doi: 10.3934/dcds.2013.33.5107 +[Abstract](436) +[PDF](526.7KB)
Abstract:
We consider the problem of the reconstruction of the source term in a parabolic Cauchy-Dirichlet system in a cylindrical domain. The supplementary information, necessary to determine the unknown part of the source term together with the solution, is given by the knowledge of an integral of the solution with respect to some of the space variables.
Well-posedness results for the Navier-Stokes equations in the rotational framework
Matthias Hieber and Sylvie Monniaux
2013, 33(11&12): 5143-5151 doi: 10.3934/dcds.2013.33.5143 +[Abstract](485) +[PDF](396.6KB)
Abstract:
Consider the Navier-Stokes equations in the rotational framework either on $\mathbb{R}^3$ or on open sets $\Omega \subset \mathbb{R}^3$ subject to Dirichlet boundary conditions. This paper discusses recent well-posedness and ill-posedness results for both situations.
Multiplicity results for classes of singular problems on an exterior domain
Eunkyoung Ko, Eun Kyoung Lee and R. Shivaji
2013, 33(11&12): 5153-5166 doi: 10.3934/dcds.2013.33.5153 +[Abstract](520) +[PDF](466.9KB)
Abstract:
We study radial positive solutions to the singular boundary value problem \begin{equation*} \begin{cases} -\Delta_p u = \lambda K(|x|)\frac{f(u)}{u^\beta} \quad \mbox{in}~ \Omega,\\ ~~~u(x) = 0 \qquad \qquad \qquad \qquad~~\mbox{if}~|x|=r_0,\\ ~~~u(x) \rightarrow 0 \qquad\qquad \qquad \mbox{if}~|x|\rightarrow \infty, \end{cases} \end{equation*} where $\Delta_p u =$ div $(|\nabla u|^{p-2}\nabla u)$, $1 < p < N, N >2, \lambda > 0, 0 \leq \beta <1 ,\Omega= \{ x \in \mathbb{R}^{N} : |x| > r_0 \}$ and $ r_0 >0$. Here $f:[0, \infty)\rightarrow (0, \infty)$ is a continuous nondecreasing function such that $\lim_{u\rightarrow \infty} \frac{f(u)}{u^{\beta+p-1}}= 0$ and $ K \in C( (r_0, \infty),(0, \infty) ) $ is such that $\int_{r_0}^{\infty} r^\mu K(r) dr < \infty, $ for some $\mu > p-1$. We establish the existence of multiple positive solutions for certain range of $\lambda$ when $f$ satisfies certain additional assumptions. A simple model that will satisfy our hypotheses is $f(u)=e^{\frac{\alpha u}{\alpha+u}}$ for $ \alpha \gg 1.$ We also extend our results to classes of systems when the nonlinearities satisfy a combined sublinear condition at infinity. We prove our results by the method of sub-super solutions.
On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators
Ismail Kombe
2013, 33(11&12): 5167-5176 doi: 10.3934/dcds.2013.33.5167 +[Abstract](420) +[PDF](399.2KB)
Abstract:
The purpose of this paper is to study the nonexistence of positive solutions of the doubly nonlinear equation \[\begin{cases} \frac{\partial u}{\partial t}=\nabla_{\gamma}\cdot (u^{m-1}|\nabla_{\gamma} u|^{p-2}\nabla_{\gamma} u) +Vu^{m+p-2} & \text{in}\quad \Omega \times (0, T ) ,\\ u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega, \\ u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T),\end{cases}\] where $\nabla_{\gamma}=(\nabla_x, |x|^{2\gamma}\nabla_y)$, $x\in \mathbb{R}^d, y\in \mathbb{R}^k$, $\gamma>0$, $\Omega$ is a metric ball in $\mathbb{R}^{N}$, $V\in L_{\text{loc}}^1(\Omega)$, $m\in \mathbb{R}$, $1 < p < d+k$ and $m + p - 2 > 0$. The exponents $q^{*}$ are found and the nonexistence results are proved for $q^{*} ≤ m+p < 3$.
Coagulation and fragmentation processes with evolving size and shape profiles: A semigroup approach
Wilson Lamb, Adam McBride and Louise Smith
2013, 33(11&12): 5177-5187 doi: 10.3934/dcds.2013.33.5177 +[Abstract](451) +[PDF](310.6KB)
Abstract:
We investigate a class of bivariate coagulation-fragmentation equations. These equations describe the evolution of a system of particles that are characterised not only by a discrete size variable but also by a shape variable which can be either discrete or continuous. Existence and uniqueness of strong solutions to the associated abstract Cauchy problems are established by using the theory of substochastic semigroups of operators.
Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system
Irena Lasiecka and Mathias Wilke
2013, 33(11&12): 5189-5202 doi: 10.3934/dcds.2013.33.5189 +[Abstract](529) +[PDF](418.0KB)
Abstract:
We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in $\mathbb{R}^n$. Global Well-posedness of solutions is shown by applying the theory of maximal parabolic regularity of type $L_p$. In addition, we prove exponential decay rates for strong solutions and their derivatives.
Stability estimates for semigroups on Banach spaces
Yuri Latushkin and Valerian Yurov
2013, 33(11&12): 5203-5216 doi: 10.3934/dcds.2013.33.5203 +[Abstract](550) +[PDF](440.3KB)
Abstract:
For a strongly continuous operator semigroup on a Banach space, we revisit a quantitative version of Datko's Theorem and the estimates for the constant $M$ satisfying the inequality $||T(t)|| ≤ M e^{\omega t}$, for all $t\ge0$, in terms of the norm of the convolution and other operators involved in Datko's Theorem. We use techniques recently developed by B. Helffer and J. Sjöstrand for the Hilbert space case to estimate $M$ in terms of the norm of the resolvent of the generator of the semigroup in the right half-plane.
Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace
Shitao Liu and Roberto Triggiani
2013, 33(11&12): 5217-5252 doi: 10.3934/dcds.2013.33.5217 +[Abstract](523) +[PDF](601.1KB)
Abstract:
We consider a second-order hyperbolic equation defined on an open bounded domain $\Omega$ in $\mathbb{R}^n$ for $n \geq 2$, with $C^2$-boundary $\Gamma = \partial \Omega = \overline{\Gamma_0 \cup \Gamma_1}$, $\Gamma_0 \cap \Gamma_1 = \emptyset$, subject to non-homogeneous Dirichlet boundary conditions for the entire boundary $\Gamma$. We then study the inverse problem of determining both the damping and the potential (source) coefficients simultaneously, in one shot, by means of an additional measurement of the Neumann boundary trace of the solution, in a suitable, explicit sub-portion $\Gamma_1$ of the boundary $\Gamma$, and over a computable time interval $T > 0$. Under sharp conditions on the complementary part $\Gamma_0 = \Gamma \backslash \Gamma_1$, $T > 0$, and under sharp regularity requirements on the data, we establish the two canonical results in inverse problems: (i) uniqueness and (ii) Lipschitz-stability. The latter (ii) is the main result of the paper. Our proof relies on a few main ingredients: (a) sharp Carleman estimates at the $H^1(\Omega) \times L^2(\Omega)$-level for second-order hyperbolic equations [23], originally introduced for control theory issues; (b) a correspondingly implied continuous observability inequality at the same energy level [23]; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Dirichlet boundary data [14,15,16]. The proof of the linear uniqueness result (Section 3) also takes advantage of a convenient tactical route ``post-Carleman estimates" proposed by V. Isakov in [8, Thm. 8.2.2, p. 231]. Expressing the final results for the nonlinear inverse problem directly in terms of the data offers an additional challenge.
An identification problem for a nonlinear one-dimensional wave equation
Alfredo Lorenzi and Eugenio Sinestrari
2013, 33(11&12): 5253-5271 doi: 10.3934/dcds.2013.33.5253 +[Abstract](469) +[PDF](388.5KB)
Abstract:
We prove the existence of a spatial coefficient in front of a nonlinear term in a one-dimensional wave equation when, in addition to classical initial and boundary condition, an integral mean involving the displacement is prescribed.
A thermo piezoelectric model: Exponential decay of the total energy
Gustavo Alberto Perla Menzala and Julian Moises Sejje Suárez
2013, 33(11&12): 5273-5292 doi: 10.3934/dcds.2013.33.5273 +[Abstract](553) +[PDF](458.3KB)
Abstract:
We consider a linear evolution model describing a piezoelectric phenomenon under thermal effects as suggested by R. Mindlin [13] and W. Nowacki [16]. We prove the equivalence between exponential decay of the total energy and an observability inequality for an anisotropic elastic wave system. Our strategy is to use a decoupling method to reduce the problem to an equivalent observability inequality for an anisotropic elastic wave system and assume a condition which guarantees that the corresponding elliptic operator has no eigenfunctions with null divergence.
How to distinguish a local semigroup from a global semigroup
J. W. Neuberger
2013, 33(11&12): 5293-5303 doi: 10.3934/dcds.2013.33.5293 +[Abstract](448) +[PDF](331.5KB)
Abstract:
For a given autonomous time-dependent system that generates either a global, in time, semigroup or else only a local, in time, semigroup, a test involving a linear eigenvalue problem is given which determines which of `global' or `local' holds. Numerical examples are given. A linear transformation $A$ is defined so that one has `global' or `local' depending on whether $A$ does not or does have a positive eigenvalue. There is a possible application to Navier-Stokes problems..
Rational approximations of semigroups without scaling and squaring
Frank Neubrander, Koray Özer and Teresa Sandmaier
2013, 33(11&12): 5305-5317 doi: 10.3934/dcds.2013.33.5305 +[Abstract](488) +[PDF](375.8KB)
Abstract:
We show that for all $q\ge 1$ and $1\le i \le q$ there exist pairwise conjugate complex numbers $b_{q,i}$ and $\lambda_{q,i}$ with $\mbox{Re} (\lambda_{q,i}) > 0$ such that for any generator $(A, D(A))$ of a bounded, strongly continuous semigroup $T(t)$ on Banach space $X$ with resolvent $R(\lambda,A) := (\lambda I-A)^{-1}$ the expression $\frac{b_{q,1}}{t}R(\frac{\lambda_{q,1}}{t},A) + \frac{b_{q,2}}{t}R(\frac{\lambda_{q,2}}{t},A) + \cdots + \frac{b_{q,q}}{t}R(\frac{\lambda_{q,q}}{t},A)$ provides an excellent approximation of the semigroup $T(t)$ on $D(A^{2q-1})$. Precise error estimates as well as applications to the numerical inversion of the Laplace transform are given.
Understanding Thomas-Fermi-Like approximations: Averaging over oscillating occupied orbitals
John P. Perdew and Adrienn Ruzsinszky
2013, 33(11&12): 5319-5325 doi: 10.3934/dcds.2013.33.5319 +[Abstract](579) +[PDF](258.5KB)
Abstract:
The Thomas-Fermi equation arises from the earliest density functional approximation for the ground-state energy of a many-electron system. Its solutions have been carefully studied by mathematicians, including J.A. Goldstein. Here we will review the approximation and its validity conditions from a physics perspective, explaining why the theory correctly describes the core electrons of an atom but fails to bind atoms to form molecules and solids. The valence electrons are poorly described in Thomas-Fermi theory, for two reasons: (1) This theory neglects the exchange-correlation energy, ``nature's glue". (2) It also makes a local density approximation for the kinetic energy, which neglects important shell-structure effects in the exact kinetic energy that are responsible for the structure of the periodic table of the elements. Finally, we present a tentative explanation for the fact that the shell-structure effects are relatively unimportant for the exact exchange energy, which can thus be more usefully described by a local density or semilocal approximation (as in the popular Kohn-Sham theory): The exact exchange energy from the occupied Kohn-Sham orbitals has an extra sum over orbital labels and an extra integration over space, in comparison to the kinetic energy, and thus averages out more of the atomic individuality of the orbital oscillations.
An interface problem: The two-layer shallow water equations
Madalina Petcu and Roger Temam
2013, 33(11&12): 5327-5345 doi: 10.3934/dcds.2013.33.5327 +[Abstract](776) +[PDF](891.0KB)
Abstract:
The aim of this article is to study a model of two superposed layers of fluid governed by the shallow water equations in space dimension one. Under some suitable hypotheses the governing equations are hyperbolic. We introduce suitable boundary conditions and establish a result of existence and uniqueness of smooth solutions for a limited time for this model.
Global existence via a multivalued operator for an Allen-Cahn-Gurtin equation
Michel Pierre and Morgan Pierre
2013, 33(11&12): 5347-5377 doi: 10.3934/dcds.2013.33.5347 +[Abstract](499) +[PDF](480.6KB)
Abstract:
The main goal of this paper is to prove existence of global solutions in time for an Allen-Cahn-Gurtin model of pseudo-parabolic type. Local solutions were known to ``blow up" in some sense in finite time. It is proved that the equation is actually governed by a monotone-like operator. It turns out to be multivalued and measure-valued. The measures are singular with respect to the Lebesgue measure. This operator allows to extend the local solutions globally in time and to fully solve the evolution problem. The asymptotic behavior is also analyzed.
Singular limits for the two-phase Stefan problem
Jan Prüss, Jürgen Saal and Gieri Simonett
2013, 33(11&12): 5379-5405 doi: 10.3934/dcds.2013.33.5379 +[Abstract](476) +[PDF](515.3KB)
Abstract:
We prove strong convergence to singular limits for a linearized fully inhomogeneous Stefan problem subject to surface tension and kinetic undercooling effects. Different combinations of $\sigma \to \sigma_0$ and $\delta\to\delta_0$, where $\sigma,\sigma_0\ge 0$ and $\delta,\delta_0\ge 0$ denote surface tension and kinetic undercooling coefficients respectively, altogether lead to five different types of singular limits. Their strong convergence is based on uniform maximal regularity estimates.
On the manifold of closed hypersurfaces in $\mathbb{R}^n$
Jan Prüss and Gieri Simonett
2013, 33(11&12): 5407-5428 doi: 10.3934/dcds.2013.33.5407 +[Abstract](462) +[PDF](406.5KB)
Abstract:
Several results from differential geometry of hypersurfaces in $\mathbb{R}^n$ are derived to form a tool box for the direct mapping method. The latter technique has been widely employed to solve problems with moving interfaces, and to study the asymptotics of the induced semiflows.
Integration with vector valued measures
M. M. Rao
2013, 33(11&12): 5429-5440 doi: 10.3934/dcds.2013.33.5429 +[Abstract](490) +[PDF](145.3KB)
Abstract:
Of the many variations of vector measures, the Fréchet variation is finite valued but only subadditive. Finding a `controlling' finite measure for these in several cases, it is possible to develop a useful integration of the Bartle-Dunford-Schwartz type for many linear metric spaces. These include the generalized Orlicz spaces, $L^{\varphi}(\mu)$, where $\varphi$ is a concave $\varphi$-function with applications to stochastic measures $Z(\cdot)$ into various Fréchet spaces useful in prediction theory. In particular, certain $p$-stable random measures and a (sub) class of these leading to positive infinitely divisible ones are detailed.
Estimating eigenvalues of an anisotropic thermal tensor from transient thermal probe measurements
Steve Rosencrans, Xuefeng Wang and Shan Zhao
2013, 33(11&12): 5441-5455 doi: 10.3934/dcds.2013.33.5441 +[Abstract](499) +[PDF](408.0KB)
Abstract:
We propose a new method for estimating the eigenvalues of the thermal tensor of an anisotropically heat-conducting material, from transient thermal probe measurements of a heated thin cylinder.
    We assume the principal axes of the thermal tensor to have been identified, and that the cylinder is oriented parallel to one of these axes (but we outline what is needed to overcome this limitation). The method involves estimating the first two Dirichlet eigenvalues (exponential decay rates) from transient thermal probe data. These implicitly determine the thermal diffusion coefficients (thermal tensor eigenvalues) in the directions of the other two axes. The process is repeated two more times with cylinders parallel to each of the remaining axes.
    The method is tested by simulating a temperature probe time-series (obtained by solving the anisotropic heat equation numerically) and comparing the computed thermal tensor eigenvalues with their true values. The results are generally accurate to less than $1\%$ error.
Hardy type inequalities and hidden energies
Juan Luis Vázquez and Nikolaos B. Zographopoulos
2013, 33(11&12): 5457-5491 doi: 10.3934/dcds.2013.33.5457 +[Abstract](458) +[PDF](557.4KB)
Abstract:
We obtain new insights into Hardy type Inequalities and the evolution problems associated to them. Surprisingly, the connection of the energy with the Hardy functionals is nontrivial, due to the presence of a Hardy singularity energy. This corresponds to a loss for the total energy. These problems are defined on bounded domains or the whole space.
    We also consider equivalent problems with inverse square potential on exterior domains or the whole space. The extra energy term is then present as an effect that comes from infinity, a kind of hidden energy. In this case, in an unexpected way, this term is additive to the total energy, and it may even constitute the main part of it.
Semi linear parabolic equations with nonlinear general Wentzell boundary conditions
Mahamadi Warma
2013, 33(11&12): 5493-5506 doi: 10.3934/dcds.2013.33.5493 +[Abstract](517) +[PDF](411.6KB)
Abstract:
Let $A$ be a uniformly elliptic operator in divergence form with bounded coefficients. We show that on a bounded domain $Ω ⊂ \mathbb{R}^N$ with Lipschitz continuous boundary $∂Ω$, a realization of $Au-\beta_1(x,u)$ in $C(\bar{Ω})$ with the nonlinear general Wentzell boundary conditions $[Au-\beta_1(x,u)]|_{∂Ω}-\Delta_\Gamma u+\partial_\nu^au+\beta_2(x,u)= 0$ on $∂Ω$ generates a strongly continuous nonlinear semigroup on $C(\bar{Ω})$. Here, $\partial_\nu^au$ is the conormal derivative of $u$, and $\beta_1(x,\cdot)$ ($x \in Ω$), $\beta_2(x,\cdot)$ ($x \in ∂Ω$) are continuous on $\mathbb{R}$ satisfying a certain growth condition.
Positive solutions of nonlinear equations via comparison with linear operators
Jeffrey R. L. Webb
2013, 33(11&12): 5507-5519 doi: 10.3934/dcds.2013.33.5507 +[Abstract](422) +[PDF](348.9KB)
Abstract:
We discuss positive solutions of problems that arise from nonlinear boundary value problems in the particular situation where the nonlinear term $f(t,u)$ depends explicitly on $t$ and this dependence is crucial. We give new fixed point index results using comparisons with linear operators. These prove new results on existence of positive solutions under some conditions which can be sharp.
An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure
Qi S. Zhang
2013, 33(11&12): 5521-5523 doi: 10.3934/dcds.2013.33.5521 +[Abstract](469) +[PDF](225.2KB)
Abstract:
We construct a global smooth solution of 3 dimensional Navier-Stokes equations in the torus, which also solves the heat equation. The solution is three dimensional and it can be arbitrarily large.
Weighted pseudo almost automorphic mild solutions to semilinear integral equations with $S^{p}$-weighted pseudo almost automorphic coefficients
Rui Zhang, Yong-Kui Chang and G. M. N'Guérékata
2013, 33(11&12): 5525-5537 doi: 10.3934/dcds.2013.33.5525 +[Abstract](683) +[PDF](406.8KB)
Abstract:
In this paper, we consider the existence of weighted pseudo almost automorphic solutions of the semilinear integral equation $x(t)= \int_{-\infty}^{t}a(t-s)[Ax(s) + f(s,x(s))]ds, \ t\in\mathbb{R}$ in a Banach space $\mathbb{X}$, where $a\in L^{1}(\mathbb{R}_{+})$, $A$ is the generator of an integral resolvent family of linear bounded operators defined on the Banach space $\mathbb{X}$, and $f : \mathbb{R}\times\mathbb{X} \rightarrow \mathbb{X}$ is a weighted pseudo almost automorphic function. The main results are proved by using integral resolvent families, suitable composition theorems combined with the theory of fixed points.

2017  Impact Factor: 1.179

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