Asymptotic Behavior for a Coupled Petrovsky–Petrovsky System with Infinite Memories

Abstract

The main goal of this article is to obtain the existence of solutions for a nonlinear system of a coupled Petrovsky–Petrovsky system in the presence of infinite memories under minimal assumptions on the functions g1,g2 and φ1,φ2. Here, g1,g2 are relaxation functions and φ1,φ2 represent the sources. Also, a general decay rate for the associated energy is established. Our work is partly motivated by recent results, with a necessary modification imposed by the nature of our problem. In this work, we limit our results to studying the system in a bounded domain. The case of the entire domain Rn requires separate consideration. Of course, obtaining such a result will require not only serious technical work but also the use of new techniques and methods. In particular, one of the most significant points in achieving this goal is the use of the perturbed Lyapunov functionals combined with the multiplier method. To the best of our knowledge, there is no result addressing the linked Petrovsky–Petrovsky system in the presence of infinite memory, and we have overcome this lacune.