Well-posedness and Stabiliy result of Petrovsky equation with a nonlinear strong damping and delay term

TL;DR

This paper investigates the well-posedness and stability of a nonlinear Petrovsky equation with delay and strong damping, establishing global existence of solutions and stability criteria using energy and convexity methods.

Contribution

It introduces new conditions for the existence and stability of solutions to a nonlinear Petrovsky equation with delay and strong damping.

Findings

Proved global existence of solutions in Sobolev spaces.

Derived stability estimates using convex function properties.

Established conditions on delay and damping weights for stability.

Abstract

In this paper we consider a nonlinear Petrovsky equation in a bounded domain with a delay term and a strong dissipation \begin{align*} u_{tt} + \Delta^{2} u -\mu_1g_1( \Delta( u_t(x,t))) -\mu_2g_2( \Delta (u_t(x,t-\tau))) =0. \end{align*} 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.