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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, whichwe 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 globalattractors, 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 isindependent of the perturbation parameter $\ep.$ The family of globalattractors are upper-semicontinuous with respect to the perturbation parameter$\ep$, a result which follows by an application of a new abstract result alsocontained in this article. Finally, we show that it is possible to obtain theglobal attractors using weaker assumptions on the nonlinear term $f$, however,in that case, the optimal regularity, the finite dimensionality, and theupper-semicontinuity of the global attractors does not necessarily hold.

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