Existence and continuity of the attractor for a singularly perturbed hyperbolic equation (Q1908803)

The author considers the semilinear damped wave equation of the form \[ \varepsilon u_{tt}+ \gamma u_t+ b(u_t)= \Delta u- f(u)+ g(x), \] where \(u= u(x, t)\), \(x\in \Omega\subset \mathbb{R}^N\) with \(\Omega\) a bounded domain. The damping function \(b\) is nondecreasing. The existence of a global compact attractor in the usual energy space \((u, u_t)\in H^1_0\times L^2\) is proved under some growth restrictions concerning the function \(f\). In particular, \(f\) is supposed to be globally Lipschitz which seems to be rather restrictive. If, in addition, \(b\) is globally Lipschitz, the upper semicontinuity of the attractors with respect to the parameter \(\varepsilon\) is shown.