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).


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