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

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 $n⩽\frac{10}{3}\sigma$. 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 $1<p⩽p_{\mathrm{crit}}(n,\sigma )$. 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.

30 pages, 1 table. Comments are welcome

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• Small data global existence for the semilinear wave equation with space-time dependent damping