Attractors for Damped Semilinear Wave Equations with Singularly Perturbed Acoustic Boundary Conditions

Abstract: Under consideration is the damped semilinear wave equation [ u_{tt}+u_t-\Delta u+u+f(u)=0 ] in a bounded domain $\Omega$ in $\mathbb{R}^3$ subject to an acoustic boundary condition with a singular perturbation, which we term "massless acoustic perturbation," [ \ep\delta_{tt}+\delta_t+\delta = -u_t\quad\text{for}\quad \ep\in[0,1]. ] By adapting earlier work by S. Frigeri, we prove the existence of a family of global attractors for each $\ep\in[0,1]$. We also establish the optimal regularity for the global attractors, as well as the existence of an exponential attractor, for each $\ep\in[0,1].$ The later result insures the global attractors possess finite (fractal) dimension, however, we cannot yet guarantee that this dimension is independent of the perturbation parameter $\ep.$ The family of global attractors are upper-semicontinuous with respect to the perturbation parameter $\ep$, a result which follows by an application of a new abstract result also contained in this article. Finally, we show that it is possible to obtain the global attractors using weaker assumptions on the nonlinear term $f$, however, in that case, the optimal regularity, the finite dimensionality, and the upper-semicontinuity of the global attractors does not necessarily hold.