general energy decay and exponential instability to a nonlinear dissipative-dispersive viscoelastic petrovsky equation

This work is concerned with the initial boundary value problem for a nonlinear viscoelastic petrovsky wave equation utt + ∆2 u − z t 0 g(t − τ )∆2 u(τ )dτ − ∆ut − ∆utt + ut|ut| m−1 = u|u| p−1 . under suitable conditions on the relaxation function g, the global existence of solutions is obtained without any relation between m and p. the uniform decay of solutions is proved by adapting the perturbed energy method. for p > m and sufficient conditions on g, an unboundedness result of solutions is also obtained.