General energy decay and exponential instability to a nonlinear dissipative-dispersive viscoelastic Petrovsky equation

Abstract

This work is concerned with the initial boundary valueproblem for a nonlinear viscoelastic Petrovsky wave equation$$u_{tt}+\Delta^{2}u-\int_{0}^{t}g(t-\tau)\Delta^{2}u(\tau)d\tau-\Delta u_{t}-\Delta u_{tt}+u_{t}|u_{t}|^{m-1}=u|u|^{p-1}.$$ Under suitable conditions on the relaxation function $g$,  the globalexistence of solutions is obtained without any relation between$m$ and $p$. The uniform decay of solutions is proved by adaptingthe perturbed energy method. For $p>m$ and sufficient conditionson $g$, an unboundedness result of solutions is also obtained.