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A non-autonomous strongly damped wave equation: existence and continuity of the pullback attractor. (English) Zbl 1213.35121

Summary: We consider the strongly damped wave equation with time-dependent terms
\[ u_{tt}-\Delta u-\gamma (t)\Delta u_t+\beta_\varepsilon(t)u_t= f(u), \]
in a bounded domain \(\Omega\subset\mathbb R^n\), under some restrictions on \(\beta_\varepsilon(t)\), \(\gamma(t)\) and growth restrictions on the nonlinear term \(f\). The function \(\beta_\varepsilon(t)\) depends on a parameter \(\varepsilon\), \(\beta_\varepsilon(t)@>\varepsilon\to0>>0\). We prove, under suitable assumptions, local and global well-posedness (using the uniform sectorial operators theory), the existence and regularity of pullback attractors \(\{{\mathcal A}_\varepsilon(t):t\in\mathbb R\}\), uniform bounds for these pullback attractors, characterization of these pullback attractors and their upper and lower semicontinuity at \(\varepsilon=0\).

MSC:

35B41 Attractors
35L71 Second-order semilinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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