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Discrete & Continuous Dynamical Systems - S

2009 , Volume 2 , Issue 1

A special issue on
Asymptotic Behavior of Dissipative PDEs

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Preface
M. Grasselli , Alain Miranville and  Sergey Zelik
2009, 2(1): i-i doi: 10.3934/dcdss.2009.2.1i +[Abstract](26) +[PDF](26.3KB)
Abstract:
This issue consists of ten carefully refereed papers dealing with important qualitative features of dissipative PDEs, with applications to fluid mechanics (compressible Navier-Stokes equations and water waves), reaction-diffusion systems ((bio)chemical reactions and population dynamics), plasma physics and phase separation and transition.
Several contributions are concerned with issues such as regularity, stability and decay rates of solutions. Furthermore, an emphasis is laid on the study of the global dynamics of the systems, in terms of attractors, and of the convergence of single trajectories to stationary solutions.
We wish to thank the referees for their valuable help in evaluating and improving the papers.
Convergence to equilibria of solutions to a conserved Phase-Field system with memory
Sergiu Aizicovici and  Hana Petzeltová
2009, 2(1): 1-16 doi: 10.3934/dcdss.2009.2.1 +[Abstract](31) +[PDF](238.1KB)
Abstract:
We show that the trajectories of a conserved phase-field model with memory are compact in the space of continuous functions and, for an exponential relaxation kernel, we establish the convergence of solutions to a single stationary state as time goes to infinity. In the latter case, we also estimate the rate of decay to equilibrium.
Non-autonomous attractors for integro-differential evolution equations
Tomás Caraballo and  P.E. Kloeden
2009, 2(1): 17-36 doi: 10.3934/dcdss.2009.2.17 +[Abstract](78) +[PDF](245.6KB)
Abstract:
We show that infinite-dimensional integro-differential equations which involve an integral of the solution over the time interval since starting can be formulated as non-autonomous delay differential equations with an infinite delay. Moreover, when conditions guaranteeing uniqueness of solutions do not hold, they generate a non-autonomous (possibly) multi-valued dynamical system (MNDS). The pullback attractors here are defined with respect to a universe of subsets of the state space with sub-exponetial growth, rather than restricted to bounded sets. The theory of non-autonomous pullback attractors is extended to such MNDS in a general setting and then applied to the original integro-differential equations. Examples based on the logistic equations with and without a diffusion term are considered.
Long-time asymptotic behavior of two-dimensional dissipative Boussinesq systems
Min Chen and  Olivier Goubet
2009, 2(1): 37-53 doi: 10.3934/dcdss.2009.2.37 +[Abstract](40) +[PDF](290.2KB)
Abstract:
In this article, we consider the two-dimensional dissipative Boussinesq systems which model surface waves in three space dimensions. The long time asymptotics of the solutions for a large class of such systems are obtained rigorously for small initial data.
The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness
Alexey Cheskidov and  Songsong Lu
2009, 2(1): 55-66 doi: 10.3934/dcdss.2009.2.55 +[Abstract](82) +[PDF](208.4KB)
Abstract:
We study the uniform global attractor for a general nonautonomous reaction-diffusion system without uniqueness using a new developed framework of an evolutionary system. We prove the existence and the structure of a weak uniform (with respect to a symbol space) global attractor $\mathcal A$. Moreover, if the external force is normal, we show that this attractor is in fact a strong uniform global attractor. The existence of a uniform (with respect to the initial time) global attractor $\mathcal A^0$ also holds in this case, but its relation to $\mathcal A$ is not yet clear due to the non-uniqueness feature of the system.
Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions
Moez Daoulatli , Irena Lasiecka and  Daniel Toundykov
2009, 2(1): 67-94 doi: 10.3934/dcdss.2009.2.67 +[Abstract](140) +[PDF](368.9KB)
Abstract:
This paper studies a wave equation on a bounded domain in $\bbR^d$ with nonlinear dissipation which is localized on a subset of the boundary. The damping is modeled by a continuous monotone function without the usual growth restrictions imposed at the origin and infinity. Under the assumption that the observability inequality is satisfied by the solution of the associated linear problem, the asymptotic decay rates of the energy functional are obtained by reducing the nonlinear PDE problem to a linear PDE and a nonlinear ODE. This approach offers a generalized framework which incorporates the results on energy decay that appeared previously in the literature; the method accommodates systems with variable coefficients in the principal elliptic part, and allows to dispense with linear restrictions on the growth of the dissipative feedback map.
A stabilizing effect of a high-frequency driving force on the motion of a viscous, compressible, and heat conducting fluid
Eduard Feireisl and  Dalibor Pražák
2009, 2(1): 95-111 doi: 10.3934/dcdss.2009.2.95 +[Abstract](33) +[PDF](237.5KB)
Abstract:
We study the impact of an oscillating external force on the motion of a viscous, compressible, and heat conducting fluid. Assuming that the frequency of oscillations increases sufficiently fast as the time goes to infinity, the solutions are shown to stabilize to a spatially homogeneous static state.
Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions
Ciprian G. Gal and  Alain Miranville
2009, 2(1): 113-147 doi: 10.3934/dcdss.2009.2.113 +[Abstract](38) +[PDF](424.0KB)
Abstract:
We consider a model of non-isothermal phase separation taking place in a confined container. The order parameter $\phi $ is governed by a viscous or non-viscous Cahn-Hilliard type equation which is coupled with a heat equation for the temperature $\theta $. The former is subject to a nonlinear dynamic boundary condition recently proposed by physicists to account for interactions with the walls, while the latter is endowed with a standard (Dirichlet, Neumann or Robin) boundary condition. We indicate by $\alpha $ the viscosity coefficient, by $\varepsilon $ a (small) relaxation parameter multiplying $\partial _{t}\theta $ in the heat equation and by $\delta $ a small latent heat coefficient (satisfying $\delta \leq \lambda \alpha $, $\delta \leq \overline{\lambda }\varepsilon $, $\lambda , \overline{\lambda }>0$) multiplying $\Delta \theta $ in the Cahn-Hilliard equation and $\partial _{t}\phi $ in the heat equation. Then, we construct a family of exponential attractors $\mathcal{M}_{\varepsilon ,\delta ,\alpha }$ which is a robust perturbation of an exponential attractor $\mathcal{M} _{0,0,\alpha }$ of the (isothermal) viscous ($\alpha >0$) Cahn-Hilliard equation, namely, the symmetric Hausdorff distance between $\mathcal{M} _{\varepsilon ,\delta ,\alpha }$ and $\mathcal{M}_{0,0,\alpha }$ goes to 0, for each fixed value of $\alpha >0,$ as $( \varepsilon ,\delta) $ goes to $(0,0),$ in an explicitly controlled way. Moreover, the robustness of this family of exponential attractors $\mathcal{M}_{\varepsilon ,\delta ,\alpha }$ with respect to $( \delta ,\alpha ) \rightarrow ( 0,0) ,$ for each fixed value of $\varepsilon >0,$ is also obtained. Finally, assuming that the nonlinearities are real analytic, with no growth restrictions, the convergence of solutions to single equilibria, as time goes to infinity, is also proved.
Finite dimensionality of a Klein-Gordon-Schrödinger type system
Marilena N. Poulou and  Nikolaos M. Stavrakakis
2009, 2(1): 149-161 doi: 10.3934/dcdss.2009.2.149 +[Abstract](29) +[PDF](216.9KB)
Abstract:
In this paper we study the finite dimensionality of the global attractor for the following system of Klein-Gordon-Schrödinger type

$ i\psi_t +\kappa \psi_{xx} +i\alpha\psi = \phi\psi+f,$
$ \phi_{tt}- \phi_{xx}+\phi+\lambda\phi_t = -Re \psi_{x}+g, $
$\psi (x,0)=\psi_0 (x), \phi(x,0) = \phi_0 (x), \phi_t (x,0)=\phi_1(x),$
$ \psi(x,t)= \phi(x,t)=0, x \in \partial \Omega, t>0, $

where $x \in \Omega, t>0, \kappa > 0, \alpha >0, \lambda >0,$ $f$ and $g$ are driving terms and $\Omega$ is a bounded interval of R With the help of the Lyapunov exponents we give an estimate of the upper bound of its Hausdorff and Fractal dimension.

Interior regularity of the compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum
Yuming Qin , Lan Huang , Shuxian Deng , Zhiyong Ma , Xiaoke Su and  Xinguang Yang
2009, 2(1): 163-192 doi: 10.3934/dcdss.2009.2.163 +[Abstract](43) +[PDF](278.5KB)
Abstract:
This paper is concerned with the interior regularity of global solutions for the one-dimensional compressible isentropic Navier-Stokes equations with degenerate viscosity coefficient and vacuum. The viscosity coefficient $\mu$ is proportional to $\rho^{\theta}$ with $0<\theta<1/3$, where $\rho$ is the density. The global existence has been established in [44] (Vong, Yang and Zhu, J. Differential Equations, 192(2), 475--501). Some ideas and more delicate estimates are introduced to prove these results.
Asymptotical dynamics of Selkov equations
Yuncheng You
2009, 2(1): 193-219 doi: 10.3934/dcdss.2009.2.193 +[Abstract](65) +[PDF](310.6KB)
Abstract:
The existence of a global attractor for the solution semiflow of Selkov equations with Neumann boundary conditions on a bounded domain in space dimension $n\le 3$ is proved. This reaction-diffusion system features the oppositely-signed nonlinear terms so that the dissipative sign-condition is not satisfied. The asymptotical compactness is shown by a new decomposition method. It is also proved that the Hausdorff dimension and fractal dimension of the global attractor are finite.

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