Asymptotic Stability for a Quasilinear Viscoelastic Equation with Nonlinear Damping and Memory

Abstract

This paper is concerned with the asymptotic behavior of a quasilinear viscoelastic equation with nonlinear damping and memory. Assuming that the kernel $\mu (s)$ satisfies $$\mu'(s)\le -k_1\mu^m(s), \ 1\le m<\frac{3}{2}$$ we establish the exponential stability result for $m=1$ and the polynomial stability result for $1