Global Existence, Finite Time Blow-up and Vacuum Isolating Phenomena for Semilinear Parabolic Equation with Conical Degeneration

Abstract

This paper is devoted to studying a semilinear parabolic equation with conical degeneration. First, we extend previous results on the vacuum isolating of solution with initial energy $J(u_0) \lt d$, where $d$ is the mountain pass level. Concretely, we obtain the explicit vacuum region, the global existence region and the blow-up region. Moreover, as far as the blow-up solution is concerned, we estimate the upper bound of the blow-up time and blow-up rate. Second, for all $p \gt 1$, we get a new sufficient condition, which demonstrates the finite time blow-up for arbitrary initial energy, and the upper bound estimate of blow-up time is obtained.