Global existence and nonexistence of semilinear wave equation with a new condition

TL;DR

This paper investigates conditions for the global existence or blow-up of solutions to a semilinear wave equation, introducing a new integral condition and potential well method to analyze initial-boundary value problems.

Contribution

It introduces a novel integral condition for the semilinear wave equation and employs potential well techniques to determine solution behaviors.

Findings

Established invariant sets and vacuum isolation of solutions.

Derived criteria for global existence and blow-up based on initial energy.

Extended understanding of wave equations with new integral conditions.

Abstract

In this paper, we consider the initial-boundary problem for semilinear wave equation with a new condition $$\alpha \int_0^{u } f(s)ds \leq uf(u) + \beta u^2 +\alpha \sigma,$$ for some positive constants $\alpha$, $\beta$, and $\sigma$, where $\beta < \frac{\lambda_1(\alpha -2)}{2}$ with $\lambda_1$ being a first eigenvalue of Laplacian. By introducing a family of potential wells, we establish the invariant sets, vacuum isolation of solutions, global existence and blow-up solutions of semilinear wave equation for initial conditions $E(0)<d$ and $E(0)=d$.