Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with internal distributed delay

Abstract

In this article we consider a nonlinear viscoelastic Petrovsky equation in a bounded domain with distributed delay

$$\begin{aligned} \begin{aligned}&|u_{t}(x,t)|^{l}u_{tt}(x,t)+\Delta ^{2}u(x,t)-\Delta u_{tt}(x,t)-\displaystyle \int _{0}^{t}h(t-\sigma )\Delta ^{2}u(x,\sigma )\,d\sigma +\mu _{1}u_{t}(x,t)\\&\quad +\int _{\tau _{1}}^{\tau _{2}}\mu _{2}(s)u_{t}(x,t-s)ds=0,\quad x\in \Omega ,\; t>0, \end{aligned} \end{aligned}$$

and prove a global solution existence result using the energy method combined with the Faedo–Galerkin approximation , under condition on the weight of the damping and the weight of distributed delay. Also we establish the exponential stability of the solution by introducing a suitable Lyapunov functional.