Global existence and blow-up of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations

TL;DR

This paper investigates a broad class of nonlocal nonlinear wave equations with two dispersive effects, establishing conditions for the existence, global behavior, and blow-up of solutions, including well-known equations like Boussinesq.

Contribution

It introduces a unified analysis framework for doubly dispersive nonlocal wave equations, proving local existence and criteria for global existence or blow-up.

Findings

Proved local existence of solutions in Sobolev spaces.

Established conditions for global existence of solutions.

Identified criteria leading to finite-time blow-up.

Abstract

This study deals with the analysis of the Cauchy problem of a general class of nonlocal nonlinear equations modeling the bi-directional propagation of dispersive waves in various contexts. The nonlocal nature of the problem is reflected by two different elliptic pseudodifferential operators acting on linear and nonlinear functions of the dependent variable, respectively. The well-known doubly dispersive nonlinear wave equation that incorporates two types of dispersive effects originated from two different dispersion operators falls into the category studied here. The class of nonlocal nonlinear wave equations also covers a variety of well-known wave equations such as various forms of the Boussinesq equation. Local existence of solutions of the Cauchy problem with initial data in suitable Sobolev spaces is proven and the conditions for global existence and finite-time blow-up of solutions are established.