Quasi-stability and attractors for a porous-elastic system with history memory

Abstract

The paper is concerned with a porous elastic problem in a past history framework. We study its long-time behavior through the corresponding autonomous dynamical system. Instead of showing the directly the system has a bounded absorbing set, we show the system is gradient system and asymptotic smoothness, and prove the existence of a global attractor, which is characterized as unstable manifold of the set of stationary solutions. We also get the quasi-stability of the system by establishing a stabilizability inequality and therefore obtain the finite fractal dimension of the global attractor.