Well-posedness and exponential stability for a plate equation with time-varying delay and past history

Abstract

This paper is concerned with a class of plate equation with past history and time-varying delay in the internal feedback $$\begin{aligned} u_{tt}+\alpha \Delta ^2 u-\int \limits ^t_{-\infty }g(t-s)\Delta ^2 u(s)\mathrm{d}s+\mu _1u_t+\mu _2u_t(t-\tau (t))+f(u)=h(x), \end{aligned}$$ defined in a bounded domain of $${\mathbb {R}}^n$$ $$(n\ge 1)$$ with some suitable initial data and boundary conditions. For arbitrary real numbers $$\mu _1$$ and $$\mu _2$$ , we proved the global well-posedness of the problem. Results on stability of energy are also proved under some restrictions on $$\mu _1$$ , $$\mu _2$$ and $$h(x)=0$$ .