On the critical exponent and sharp lifespan estimates for semilinear damped wave equations with data from Sobolev spaces of negative order

TL;DR

This paper introduces a new critical exponent for semilinear damped wave equations with initial data in negative Sobolev spaces, establishing conditions for global existence and finite-time blow-up, and providing sharp lifespan estimates.

Contribution

It derives a novel critical exponent for these equations and analyzes solution lifespan with sharp bounds, extending understanding to negative Sobolev initial data.

Findings

Global existence for supercritical exponents

Finite-time blow-up for subcritical exponents

Sharp lifespan estimates in the subcritical case

Abstract

We study semilinear damped wave equations with power nonlinearity $|u|^p$ and initial data belonging to Sobolev spaces of negative order $\dot{H}^{-\gamma}$. In the present paper, we obtain a new critical exponent $p=p_{\mathrm{crit}}(n,\gamma):=1+\frac{4}{n+2\gamma}$ for some $\gamma\in(0,\frac{n}{2})$ and low dimensions in the framework of Soblev spaces of negative order. Precisely, global (in time) existence of small data Sobolev solutions of lower regularity is proved for $p>p_{\mathrm{crit}}(n,\gamma)$, and blow-up of weak solutions in finite time even for small data if $1<p<p_{\mathrm{crit}}(n,\gamma)$. Furthermore, in order to more accurately describe the blow-up time, we investigate sharp upper bound and lower bound estimates for the lifespan in the subcritical case.