Dynamics of wave equations with moving boundary

This paper is concerned with long-time dynamics of weakly damped semilinear wave equations defined on domains with moving boundary. Since the boundary is a function of the time variable the problem is intrinsically non-autonomous. Under the hypothesis that the lateral boundary is time-like, the solution operator of the problem generates an evolution process U(t, τ ) : Xτ → Xt, where Xt are timedependent Sobolev spaces. Then, by assuming the domains are expanding, we establish the existence of minimal pullback attractors with respect to a universe of tempered sets defined by the forcing terms. Our assumptions allow nonlinear perturbations with critical growth and unbounded time-dependent external forces.Conselho Nacional de Desenvolvimento Científico e TecnológicoMinisterio de EducaciónMinisterio de Ciencia e Innovació

New Trends on Nonlocal and Functional Boundary Value Problems

In the last decades, boundary value problems with nonlocal and functional boundary conditions have become a rapidly growing area of research. The study of this type of problems not only has a theoretical interest that includes a huge variety of differential, integrodifferential, and abstract equations, but also is motivated by the fact that these problems can be used as a model for several phenomena in engineering, physics, and life sciences that standard boundary conditions cannot describe. In this framework, fall problems with feedback controls, such as the steady states of a thermostat, where a controller at one of its ends adds or removes heat depending upon the temperature registered in another point, or phenomena with functional dependence in the equation and/or in the boundary conditions, with delays or advances, maximum or minimum arguments, such as beams where the maximum (minimum) of the deflection is attained in some interior or endpoint of the beam. Topological and functional analysis tools, for example, degree theory, fixed point theorems, or variational principles, have played a key role in the developing of this subject. This volume contains a variety of contributions within this area of research. The articles deal with second and higher order boundary value problems with nonlocal and functional conditions for ordinary, impulsive, partial, and fractional differential equations on bounded and unbounded domains. In the contributions, existence, uniqueness, and asymptotic behaviour of solutions are considered by using several methods as fixed point theorems, spectral analysis, and oscillation theory

ATTRACTORS FOR SEMILINEAR WAVE EQUATIONS WITH LOCALIZED DAMPING AND EXTERNAL FORCES

This paper is concerned with long-time dynamics of semilinear wave equations defined on bounded domains of R3 with cubic nonlinear terms and locally distributed damping. The existence of regular finite-dimensional global attractors established by Chueshov, Lasiecka and Toundykov (2008) reflects a good deal of the current state of the art on this matter. Our con tribution is threefold. First, we prove uniform boundedness of attractors with respect to a forcing parameter. Then, we study the continuity of attractors with respect to the parameter in a residual dense set. Finally, we show the exis tence of generalized exponential attractors. These aspects were not previously considered for wave equations with localized damping

Longtime Dynamics of a Semilinear Lamé System

This paper is concerned with longtime dynamics of semilinear Lamé systems ∂2 t u − μ u − (λ + μ)∇divu + α∂tu + f (u) = b, defined in bounded domains ofR3 with Dirichlet boundary condition. Firstly,we establish the existence of finite dimensional global attractors subjected to a critical forcing f (u).Writing λ + μ as a positive parameter ε, we discuss some physical aspects of the limit case ε → 0. Then, we show the upper-semicontinuity of attractors with respect to the parameter when ε → 0. To our best knowledge, the analysis of attractors for dynamics of Lamé systems has not been studied before