Optimal bounds and blow-up criteria for a semi-linear accretive wave equation

TL;DR

This paper establishes optimal bounds and blow-up criteria for solutions to a semi-linear wave equation with specific nonlinearities, providing insights into the conditions leading to finite-time blow-up.

Contribution

It introduces new energetic criteria and optimal lower bounds for blow-up solutions in a semi-linear wave equation with particular nonlinear terms.

Findings

Derived energetic blow-up criteria

Established optimal lower bounds for blow-up solutions

Identified parameter ranges for solution blow-up

Abstract

In this paper we consider the semi-linear wave equation: $u_{tt}-\Delta u=u_t|u_t|^{p-1}$ in $\mathbb{R}^N$ where $1<p\leq 1+\frac2{N-1}$ and $p<5$ if N=1, $p\neq 3$ if N=2. We give an energetic criteria and optimal lower bound for blowing up solutions of this equation.