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

Hicham Saber, Mohamed Ferhat, Amin Benaissa Cherif, Tayeb Blouhi, Ahmed Himadan (), Tariq Alraqad and Abdelkader Moumen
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Hicham Saber: Department of Mathematics, Faculty of Sciences, University of Ha’il, Ha’il 55473, Saudi Arabia
Mohamed Ferhat: Department of Mathematics, Faculty of Mathematics and Informatics, University of Science and Technology of Oran Mohamed-Boudiaf (USTOMB), El Mnaouar, BP 1505, Bir El Djir, Oran 31000, Algeria
Amin Benaissa Cherif: Department of Mathematics, Faculty of Mathematics and Informatics, University of Science and Technology of Oran Mohamed-Boudiaf (USTOMB), El Mnaouar, BP 1505, Bir El Djir, Oran 31000, Algeria
Tayeb Blouhi: Department of Mathematics, Faculty of Mathematics and Informatics, University of Science and Technology of Oran Mohamed-Boudiaf (USTOMB), El Mnaouar, BP 1505, Bir El Djir, Oran 31000, Algeria
Ahmed Himadan: Department of Mathematics, College of Sciences and Arts, Qassim University, Ar-Rass 51452, Saudi Arabia
Tariq Alraqad: Department of Mathematics, Faculty of Sciences, University of Ha’il, Ha’il 55473, Saudi Arabia
Abdelkader Moumen: Department of Mathematics, Faculty of Sciences, University of Ha’il, Ha’il 55473, Saudi Arabia

Mathematics, 2023, vol. 11, issue 21, 1-16

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 g 1 , g 2 and φ 1 , φ 2 . Here, g 1 , g 2 are relaxation functions and φ 1 , φ 2 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 R n 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.

Keywords: Lyapunov functions; energy decay; infinite memories; source terms; partial differential equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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