Critical exponent for the bipolar blowup in a semilinear parabolic equation
type: Item
instance of: scholarly article
title: Critical exponent for the bipolar blowup in a semilinear parabolic equation
zbMATH Open document ID: 0914.35064
author: Noriko Mizoguchi
author: Hirokazu Ninomiya
author: Eiji Yanagida
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-06-21T00:00:00Z
review text: The blowup of solutions of the initial-boundary value problem: \[ u_t=\Delta u+| u|^{p-1}u\quad\text{in }\Omega\times(0,\infty) \] \[ u(x,t)=0\quad \text{on }\partial\Omega\times (0,\infty),\qquad u(x,0)=u_0(x) \quad\text{in }\Omega \] is studied, where \(\Omega\) is a cone in \(\mathbb{R}^N\). A solution exhibits bipolar blowup if \[ \sup_{x\in \Omega}u(x,t) \to+\infty \quad \text{and}\quad \inf_{x\in\Omega} u(x,t) \to-\infty \] for some \(t\to\infty\). The authors show that there exists a critical exponent \(p^x>1\) for the bipolar blowup in the following sense: if \(1<p\leq p^x\), then there exist arbitrarily small initial data such that the solution exhibits the bipolar blowup, whereas if \(p>p^*\), then the bipolar blowup does not occur for any sufficiently small initial data. The value of \(p^*\) is expressed in terms of the dimension \(N\) and the second Dirichlet eigenvalue of the Laplace-Beltrami operator on \(\Omega \times S^{N-1}\). In the case of \(\Omega=R\) \(k\)-bipolar blowup is also defined, and critical exponents for the \(k\)-polar blowup are determinated.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1130513
zbMATH Keywords: semilinear parabolic equation
zbMATH Keywords: small initial data
reviewed by: Maria Agostina Vivaldi
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5815
OpenAlex ID: W2027307580
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DOI: 10.1006/JMAA.1997.5815
description: scientific article; zbMATH DE number 1130513