Thresholds for global existence and blow-up in a general class of doubly dispersive nonlocal wave equations

TL;DR

This paper investigates conditions under which solutions to a broad class of nonlocal nonlinear wave equations either exist globally or blow up, extending previous results and encompassing well-known dispersive wave equations.

Contribution

It establishes new thresholds for global existence and blow-up in nonlocal wave equations using the potential well method, improving upon existing results.

Findings

Derived new criteria for global solutions versus blow-up

Unified analysis covering various well-known wave equations

Enhanced understanding of nonlocal dispersive wave dynamics

Abstract

In this article we study global existence and blow-up of solutions for a general class of nonlocal nonlinear wave equations with power-type nonlinearities, $u_{tt}-Lu_{xx}=B(- |u|^{p-1}u)_{xx}, ~(p>1)$, where the nonlocality enters through two pseudo-differential operators $L$ and $B$. We establish thresholds for global existence versus blow-up using the potential well method which relies essentially on the ideas suggested by Payne and Sattinger. Our results improve the global existence and blow-up results given in the literature for the present class of nonlocal nonlinear wave equations and cover those given for many well-known nonlinear dispersive wave equations such as the so-called double-dispersion equation and the traditional Boussinesq-type equations, as special cases.