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) : X τ → X t , where X t 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.
In this work we study a weakly dissipative wave equation defined in domains with moving boundary ∂ 2 u ∂t 2 + η ∂ u ∂t − ∆u + g(u) = f (x,t), (x,t) ∈ Q τ , where Q τ = t∈(τ,+∞) O t × {t}. We says that a domain Q τ has moving boundary if the boundary Γ t of O t varies with respect to t. Our contribution is threefold. 1-We prove that the wave equation equipped with Dirichlet boundary condition is well-posed in the sense of Hadamard (global existence, uniqueness and continuous dependence with respect to data) for weak and strong solutions. This is done by using a classical argument that transforms the time dependent domain in a fixed domain. As a consequence we see that the problem is essentially non-autonomous. 2-We find a theory of non-autonomous dynamical systems in order to study the solution operator as a process U(t, τ) : X τ −→ X t , t ≥ τ, defined in time dependent phase spaces X t = H 1 0 (O t) × L 2 (O t). 3-In the context of long-time behavior of solutions we find suitable conditions to guarantee the existence of a pullback attractor. Roughly speaking, we assume the domain Q is expanding and time-like. We emphasize that our work is the first one that consider wave equations in noncylindrical domains as non-autonomous dynamical systems. With respect to parabolic equations, similar results were early obtained by
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