Attractors for processes on time-dependent spaces. Applications to wave equations.(English)Zbl 1288.35098

In this article the authors consider a two-parameter family of operators $$U(t, \tau):X_{\tau}\to X_t$$, acting on a one-parameter family of normed spaces. They are given conditions for $$U(t, \tau)$$ which ensure the existence and uniqueness of the time-dependent attractor $${\mathcal R}=\{A_t\}_{t\in \mathbb R}$$, where every $$A_t\subset X_t$$ attracts all solutions of the system originating sufficiently far in the past. Also, the authors give conditions for the family $$U(t, \tau)$$ which ensure the invariance of $${\mathcal R}$$.
As an application, the authors consider the following initial-boundary value problem $\begin{cases} \epsilon u_{tt}+\alpha u_t-\Delta u+f(u)=g,\quad t>\tau,\\ u_{|_{\partial\Omega}}=0,\\ u(x, \tau)=a(x),\quad u_t(x, \tau)=b(x), \end{cases}\tag{1}$ where $$a, b:\Omega\to \mathbb R$$ are assigned data, $$\epsilon=\epsilon(t)$$ is a function of $$t$$, $$\Omega\subset \mathbb R^3$$ is a bounded domain with smooth boundary $$\partial\Omega$$, $$u=u(x, t):\Omega\times [\tau, \infty)\to \mathbb R$$ is the unknown, the damping coefficient $$\alpha$$ is a positive constant.
The authors prove existence of a time dependent attractor of optimal regularity for the problem (1).

MSC:

 35B41 Attractors 35L71 Second-order semilinear hyperbolic equations 35L20 Initial-boundary value problems for second-order hyperbolic equations
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