Taiwanese Journal of Mathematics

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

Guangyu Xu

Full-text: Open access

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.

Article information

Source
Taiwanese J. Math., Volume 22, Number 6 (2018), 1479-1508.

Dates
Received: 30 October 2017
Accepted: 11 March 2018
First available in Project Euclid: 23 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1521792085

Digital Object Identifier
doi:10.11650/tjm/180302

Mathematical Reviews number (MathSciNet)
MR3878578

Zentralblatt MATH identifier
07021701

Subjects
Primary: 35K61: Nonlinear initial-boundary value problems for nonlinear parabolic equations 35B44: Blow-up 35K10: Second-order parabolic equations 35K55: Nonlinear parabolic equations 35D30: Weak solutions

Keywords
cone Sobolev space vacuum isolating phenomena global existence finite time blow-up blow-up time blow-up rate

Citation

Xu, Guangyu. Global Existence, Finite Time Blow-up and Vacuum Isolating Phenomena for Semilinear Parabolic Equation with Conical Degeneration. Taiwanese J. Math. 22 (2018), no. 6, 1479--1508. doi:10.11650/tjm/180302. https://projecteuclid.org/euclid.twjm/1521792085


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