Singular limits of binary mixtures in solids theory

Abstract

We investigate the long-term behavior, as a certain coupling parameter α becomes large, of a one-dimensional model of a binary mixture of solids with nonlinear damping and nonlinear feedback forces. We prove the existence of a smooth global attractor with finite fractal dimension and study its limiting properties when α tends to infinity. More precisely, we prove that this limit coincides with a one-dimensional single wave equation. Finally, we also prove convergence of the global attractor of the binary mixture model to the global attractor of the single wave equation as α→∞. To the best of our knowledge, this result is new for singularly perturbed systems of a binary mixture of solids.