Observability and global attractors for viscoelastic equation with degenerate memory

Abstract

This paper is concerned with the nonlinear viscoelastic equation (see equation (1)) under arbitrary degenerate kernel, nonlinear friction damping, and nonlinear source term. We analyze the influence of observability inequalities on the dynamics of the solutions of equation (1). The problem is degenerate in the sense that the function in the memory term $a(x)>0$ is allowed to vanish in a part of the domain $\overline{\mathrm{\Omega }}$ . In the present work, we consider the degenerate case by adding a complementary frictional damping $\mathrm{bh}$ , which is in a certain sense arbitrarily small, such that $\underset{x\in \overline{\mathrm{\Omega }}}{\inf }\{a(x)+b(x)\}>0$ . Moreover, by assuming a minimal condition on g and without assuming that the dissipation functions a and b satisfy a geometric control condition, we prove that the problem studied has an absorbing set. The existence of global attractor with finite fractal dimensions is obtained from observability inequalities and by the useful properties of asymptotic smoothness and quasi-stability (the latter requires a somewhat stronger condition on h).