The blow-up rate of solutions of semilinear heat equations
type: Item
instance of: scholarly article
title: The blow-up rate of solutions of semilinear heat equations
zbMATH Open document ID: 0672.35035
DOI: 10.1016/0022-0396(89)90159-9
author: W. Liu
published in: Journal of Differential Equations
publication date: 1989-01-01T00:00:00Z
review text: This paper deals with the asymptotic behavior of the solution of the semilinear equation \[ u_ t-\Delta u=e^ u\quad in\quad \Omega \times (0,T),\quad u=-K\leq 0\quad on\quad \partial \Omega \times (0,T), \] \[ u(x,0)=\phi (x)\geq -K\quad for\quad x\in \Omega \subset {\mathbb{R}}^ n, \] near the blow up points as \(t\to T\). The main result (Theorem 5.4) states that this behavior is of the logarithmic type. More precisely, let \(\Omega\) be a convex domain and \(\Delta \phi +e^{\phi}\geq 0.\) If \(n\leq 2\) and a is a blow-up point (i.e. there exists a sequence \(\{(x_ m,t_ m)\}\) such that \(x_ m\to a\), \(t_ m\to T\) and \(u(x_ m,t_ m)\to +\infty)\), then u(x,t)-log(1/(T-t))\(\to 0\) as \(t\to T\), uniformly for \(| x-a| \leq C(T-t)^{1/2},\) where C is any positive constant. The author also proves (for any dimension n) the nondegeneracy of any blow-up point. In the case of the Cauchy problem \((\Omega ={\mathbb{R}}^ n)\), the blow-up set is bounded, provided \(\phi\) satisfies some growth restrictions at \(\infty\). In order to prove the main result, an energy estimate and a Pohozaev's type identity are also established. Note that analogous results for the semilinear heat equation \(u_ t-\Delta u=| u|^{p-1}u\) was obtained by \textit{Y. Giga} and \textit{R. V. Kohn} [Commun. Pure Appl. Math. 38, 297-319 (1985; Zbl 0585.35051) and Indiana Univ. Math. J. 36, 1-40 (1987; Zbl 0601.35052)].
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 4099791
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: semilinear
zbMATH Keywords: blow up
zbMATH Keywords: Cauchy problem
zbMATH Keywords: energy estimate
zbMATH Keywords: Pohozaev's type identity
MaRDI profile type: Publication
full work available at URL: 0022-0396(89)90159-9
OpenAlex ID: W2005028941
cites work: A Mathematical Analysis of Blowup for Thermal Reactions—the Spatially Nonhomogeneous Case
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cites work: Nondegeneracy of blowup for semilinear heat equations
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description: scientific article; zbMATH DE number 4099791