## 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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