Solvability for a New Class of Moore-Gibson-Thompson Equation with Viscoelastic Memory, Source Terms, and Integral Condition

Salah Mahmoud Boulaaras,Mohamed Abdalla,Abdelbaki Choucha,Djamel Ouchenane,Bahri Belkacem Cherif,Asma Alharbi,Santosh Kumar

doi:10.1155/2021/9932354

Abstract

This paper deals with the existence and uniqueness of solutions of a new class of Moore-Gibson-Thompson equation with respect to the nonlocal mixed boundary value problem, source term, and nonnegative memory kernel. Galerkin’s method was the main used tool for proving our result. This work is a generalization of recent homogenous work.

Highlights

In this contribution, we are interested to study the existence and uniqueness of solutions of the following problem 8 >>>>>>< Luðx, tÞ = auttt + βutt −c2Δu bΔut ðt hðt σÞΔuðσÞdσ Fðx, tÞ, >>>>>>: uðx, 0Þ
As the existence of the third derivative is very important, especially in the field of thermodynamics (EIT), the study of these models is considered the beginning of an in-depth understanding of both convergent and good behavior
We find a system of differential equations of fifth order with respect to t, constant coefficients, and the initial conditions (21)

Summary

Introduction

We are interested to study the existence and uniqueness of solutions of the following problem. As the existence of the third derivative is very important, especially in the field of thermodynamics (EIT), the study of these models is considered the beginning of an in-depth understanding of both convergent and good behavior. Nonlinear problems of great importance can be considered [9], where Galerkin’s method was applied in solving them, for more depth ([2, 3, 10,11,12,13]). In [1], the authors used Galerkin’s method to demonstrate the ability to solve a mixed problem of MGT equation in the absence of viscous elasticity and memory. We apply Galerkin’s method to prove the existence, and in the fourth part, we demonstrate the uniqueness

Preliminaries

Uniqueness of the Problem

Conclusion