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 (−∆) σ and power nonlinearity |u| p , which was proposed by Bezerra-Carvalho-Santos [2] recently. In this manuscript, we obtain a new critical exponent p = pcrit(n, σ) := 1 + 6σ max{3n−4σ,0} for n 10 3 σ. Precisely, the global (in time) existence of small data Sobolev solutions is proved for the supercritical case p > pcrit(n, σ), and weak solutions blow up in finite time even for small data if 1 < p pcrit (n, σ). 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.