Finite dimensional attractor for a composite system of wave/plate equations with localised damping

The long-term behaviour of solutions to a model for acoustic-structure interactions is addressed; the system is comprised of coupled semilinear wave (3D) and plate equations with nonlinear damping and critical sources. The questions of interest are: existence of a global attractor for the dynamics generated by this composite system, as well as dimensionality and regularity of the attractor. A distinct and challenging feature of the problem is the geometrically restricted dissipation on the wave component of the system. It is shown that the existence of a global attractor of finite fractal dimension -- established in a previous work by Bucci, Chueshov and Lasiecka (Comm. Pure Appl. Anal., 2007) only in the presence of full interior acoustic damping -- holds even in the case of localised dissipation. This nontrivial generalization is inspired by and consistent with the recent advances in the study of wave equations with nonlinear localised damping.Comment: 40 pages, 1 figure; v2: added references for Section 1, submitte

Decay rates for solutions of semilinear wave equations with a memory condition at the boundary

In this paper, we study the stability of solutions for semilinear wave equations whoseboundary condition includes a integral that represents the memory effect. We show that the dissipation is strong enough to produce exponential decay of the solution, provided the relaxation function also decays exponentially. When the relaxation function decays polinomially, we show that the solution decays polynomially and with the same rate