Local and Global Existence of Solution for Love Type Waves with Past History

Mohamed Biomy,Khaled Zennir,Ahmed Himadan

doi:10.3390/math8111998

Abstract

In this paper, we consider an initial boundary value problem for nonlinear Love equation with infinite memory. By combining the linearization method, the Faedo–Galerkin method, and the weak compactness method, the local existence and uniqueness of weak solution is proved. Using the potential well method, it is shown that the solution for a class of Love-equation exists globally under some conditions on the initial datum and kernel function.

Highlights

Love equation is a one-dimensional mathematical model that is used to determine a many physical phenomenon
An initial boundary value problem for a nonlinear Love equation with infinite memory has been considered by Zennir and et al in [4] and the finite time blow up of weak solution has been shown under a relationship between the relaxation function g and nonlinear sources, i.e., kuk → ∞ when t → T ∗ (T ∗ is a finite time)
In order to complete the study, we have to address the problem in quantitative terms. This is the subject of our present article, from a different angle, where we proved in detail, with the use of the most modern methods, the local existence and global existence of solution on (0, ∞)

Summary

Introduction

Love equation is a one-dimensional mathematical model that is used to determine a many physical phenomenon. − ( G ( x, t, y, y x , ∂t y, ∂t y x )) x + f ( x, t), with initial conditions and homogeneous Dirichlet boundary conditions and the authors established the existence of a unique local weak solution, a blow-up result for solutions with negative initial energy, the global existence and exponential decay of weak solution. ∂tt y − (μ( x, t)y x ) x + f (y, ∂t y) = F ( x, t), with initial conditions and boundary conditions of two-point type and the authors proved existence of a weak solution, uniqueness, regularity, and decay properties of solution. + f ( x, t), with initial conditions and homogeneous Dirichlet boundary conditions and the authors proved existence and uniqueness of a solution, a blow-up of the solution with a negative initial energy and the exponential decay of weak solution. Love-equation exists globally under some conditions on the initial datum

Existence and Uniqueness of Local Weak Solution

Global Solution

Conclusions