Asymptotic stability of solutions to a semilinear viscoelastic equation with analytic nonlinearity

Abstract

By a small perturbation of the usual Lyapunov energy and by using the Łojasiewicz–Simon inequality, we show that the dissipation given by the memory term is strong enough to prove the convergence to equilibrium as well as estimates for the rate of convergence for any global bounded solution to the semilinear viscoelastic equation

$$\begin{aligned} |u_t |^\rho u_{tt} -\Delta u_{tt}-\Delta u +\int ^\tau _0 k(s) \Delta u(t-s)\mathrm{d}s+ f(x,u)=g, \ \tau \in \{t, \infty \}, \end{aligned}$$

in a bounded regular domain of ${\mathbb {R}}^n$ with Dirichlet boundary conditions. Here, the kernel function $k > 0$ is assumed to decay exponentially at infinity, the nonlinearity f is analytic in the second variable, and the forcing term g is supposed to decay polynomially or exponentially at infinity. The present work extends the previous result where the given equation was studied in the presence of an additional strong linear damping $-\,\Delta u_t$.