We study the effects of large diffusivity in all parts of the domain in a linearly damped wave equation subject to standard zero Robin-type boundary conditions. In the linear case, we show in a given sense that the asymptotic behaviour of solutions verifies a second-order ordinary differential equation. In the semilinear case, under suitable dissipative assumptions on the nonlinear term, we prove the existence of a global attractor for fixed diffusion and that the limiting attractor for large diffusion is finite dimensional.
Robert Willie. "Large diffusivity finite-dimensional asymptotic behaviour of a semilinear wave equation." J. Appl. Math. 2003 (8) 409 - 427, 29 June 2003. https://doi.org/10.1155/S1110757X03212067