About Blow up of Solutions With Arbitrary Positive Initial Energy to Nonlinear Wave Equations

TL;DR

This paper investigates conditions under which solutions to certain nonlinear wave equations with positive initial energy blow up in finite time, extending previous results and applying to various specific equations and boundary conditions.

Contribution

It generalizes Levine's blow-up results to a broader class of nonlinear wave equations with positive initial energy, including boundary value problems.

Findings

Blow-up occurs for solutions with arbitrary positive initial energy.

Results apply to nonlinear Klein-Gordon, Boussinesq, and plate equations.

Includes blow-up results for wave equations with nonlinear boundary conditions.

Abstract

We show that blow up of solutions with arbitrary positive initial energy of the Cauchy problem for the abstract wacve eqation of the form $Pu_{tt}+Au=F(u) \ (*)$ in a Hilbert space, where $P,A$ are positive linear operators and $F(\cdot)$ is a continuously differentiable gradient operator can be obtained from the result of H.A. Levine on the growth of solutions of the Cauchy problem for (*). This result is applied to the study of inital boundary value problems for nonlinear Klein-Gordon equations, generalized Boussinesq equations and nonlinear plate equations. A result on blow up of solutions with positive initial energy of the initial boundary value problem for wave equation under nonlinear boundary condition is also obtained.