Blow-up rate for a semilinear wave equation with exponential nonlinearity in one space dimension

TL;DR

This paper investigates the blow-up behavior of solutions to a one-dimensional semilinear wave equation with exponential nonlinearity, establishing blow-up rates under less restrictive initial data conditions than previous studies.

Contribution

It extends previous results by deriving blow-up rates with lower regularity initial data, broadening understanding of solution behavior in semilinear wave equations.

Findings

Derived blow-up rates near non-characteristic points.

Provided bounds on blow-up behavior at other points.

Generalized prior results to less regular initial data.

Abstract

We consider in this paper blow-up solutions of the semilinear wave equation in one space dimension, with an exponential source term. Assuming that initial data are in $H^{1}_{loc}\times L^2_{loc}$ or some times in $ W^{1,\infty}\times L^{\infty}$, we derive the blow-up rate near a non-characteristic point in the smaller space, and give some bounds near other points. Our result generalize those proved by Godin under high regularity assumptions on initial data.