Critical exponent and sharp lifespan estimates for semilinear third-order evolution equations

TL;DR

This paper establishes a new critical exponent for third-order evolution equations with fractional Laplacian, determining conditions for global existence or finite-time blow-up of solutions, and provides sharp lifespan estimates.

Contribution

It introduces a novel critical exponent for these equations and derives precise lifespan bounds, advancing understanding of solution behavior in different regimes.

Findings

Global existence for supercritical p

Finite-time blow-up for subcritical p

Sharp lifespan estimates in critical case

Abstract

We study semilinear third-order (in time) evolution equations with fractional Laplacian $(-\Delta)^{\sigma}$ and power nonlinearity $|u|^p$, which was proposed by Bezerra-Carvalho-Santos [2] recently. In this manuscript, we obtain a new critical exponent $p=p_{\mathrm{crit}}(n,\sigma):=1+\frac{6\sigma}{\max\{3n-4\sigma,0\}}$ for $n\leqslant\frac{10}{3}\sigma$. Precisely, the global (in time) existence of small data Sobolev solutions is proved for the supercritical case $p>p_{\mathrm{crit}}(n,\sigma)$, and weak solutions blow up in finite time even for small data if $1<p\leqslant p_{\mathrm{crit}}(n,\sigma)$. Furthermore, to more accurately describe the blow-up time, we derive new and sharp upper bound as well as lower bound estimates for the lifespan in the subcritical case and the critical case.