Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations

TL;DR

This paper investigates a broad class of nonlinear nonlocal coupled wave equations, establishing conditions for both finite-time blow-up and global existence of solutions, with applications to elasticity and lattice models.

Contribution

It introduces a general framework for analyzing nonlocal coupled wave equations, providing new criteria for solution blow-up and global existence.

Findings

Conditions for finite-time blow-up of solutions.

Criteria for global existence of solutions.

Application to elasticity and lattice models.

Abstract

We study the initial-value problem for a general class of nonlinear nonlocal coupled wave equations. The problem involves convolution operators with kernel functions whose Fourier transforms are nonnegative. Some well-known examples of nonlinear wave equations, such as coupled Boussinesq-type equations arising in elasticity and in quasi-continuum approximation of dense lattices, follow from the present model for suitable choices of the kernel functions. We establish local existence and sufficient conditions for finite time blow-up and as well as global existence of solutions of the problem.