Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with a delay term in the non-linear internal feedback

Abstract

In this article we consider a nonlinear viscoelastic Petrovsky equation in a bounded domain with a delay term in the weakly nonlinear internal feedback: $$\eqalign{ &|u_t|^{l}u_{tt} +\Delta^2 u -\Delta u_{tt} -\int_0^t h(t-s)\Delta^2 u(s)\,ds\cr &+\mu_1g_1(u_t(x,t)) +\mu_2g_2(u_t(x,t-\tau))=0. }$$ We prove the existence of global solutions in suitable Sobolev spaces by using the energy method combined with Faedo-Galarkin method under condition on the weight of the delay term in the feedback and the weight of the term without delay. Furthermore, we study general stability estimates by using some properties of convex functions.

Keywords

• Global solution
• delay term
• general decay
• multiplier method
• weak frictional damping
• convexity
• viscoelastic Petrovsky equation