The Long-Time Stability of Solutions for Intermittently Controlled Viscoelastic Wave Equations with Memory Terms

Abstract

In this paper, we investigate the asymptotic stability of global solutions to the following intermittently controlled semilinear viscoelastic equation with memory term $$\begin{aligned}u(x,t)\in \varOmega \times [0,\infty ), \end{aligned}$$ under the null Dirichlet boundary condition and $$\tau \in \{t,\infty \}$$ . By virtue of appropriate new Lyapunov functional and Łojasiewicz–Simon inequality, we show that any global bounded solution converges to a steady state and get the rate of convergence as well, when damping coefficients are integral positive and positive-negative, respectively. Moreover, under the assumptions of on–off or sign-changing dampings, we derive the asymptotic stability of solutions.