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

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.

References in corpus (3)

• Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity
• Decay estimates for the linear damped wave equation on the Heisenberg group
• On the blow-up of solutions to semilinear damped wave equations with power nonlinearity in compact Lie groups