Blow-up problems for a semilinear heat equation with large diffusion
type: Item
instance of: scholarly article
title: Blow-up problems for a semilinear heat equation with large diffusion
zbMATH Open document ID: 1072.35096
author: Kazuhiro Ishige
author: Hiroki Yagisita
published in: Journal of Differential Equations
publication date: 2005-05-04T00:00:00Z
review text: The paper deals with the Cauchy-Neumann problem \[ \partial_t u = D \Delta u + u^p \quad \text{ for } x \in \Omega, \; 0<t<T_D, \] \[ \partial_{\nu} u(x,t) = 0 \quad \text{ for } x \in \partial \Omega, \; 0<t<T_D, \qquad u(x,0) = \phi(x) \quad \text{ for } x \in \Omega, \] where \(p>1\), \(\Omega\) is a bounded smooth domain in \(\mathbb R^N\) and \(\nu\) is the exterior unit normal to \(\partial \Omega\). Denoting by \(u_D(x,t)\) the solution corresponding to the fixed diffusive constant \(D>0\), the authors define the blow up time \(T_D\) as \[ T_D = \sup_{\tau >0}\{ u_D \in L^{\infty} ( \Omega \times (0,\tau))\} \] and the blow up set \(B_D\) as \[ \begin{multlined} B_D = \{ x \in \bar \Omega: \lim_{k \to \infty} (x_k, t_k) = (x, T_D), \\ \lim_{k \to \infty} u_D (x_k, t_k) = \infty \text{ for some sequence } (x_k, t_k) \in \bar \Omega \times (0,T_D) \}.\end{multlined} \] The function \[ u^*_D (x) = \lim_{t \nearrow T_D} u_D(x,t) \quad \text{for } x \in \bar \Omega \setminus B_D \] is called the blow up profile for the solution \(u_D\). The aim of the paper is to study the blow up time \(T_D\), the location of the blow up set \(B_D\) and the blow up profile \(u^*_D\) for sufficiently large \(D\). The paper is related to the earlier papers [\textit{H. Fujita}, J. Fac. Sci., Univ. Tokyo, Sect. I 13, 109--124 (1966; Zbl 0163.34002)] and [\textit{K. Ishige}, Adv. Differ. Equ. 7, No. 8, 1003--1024 (2002; Zbl 1036.35096)] quoted by the authors.
reviewed by: Hanna Marcinkowska
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 2164825
zbMATH Keywords: nonlinear diffusion equation
zbMATH Keywords: blow up time
zbMATH Keywords: blow up set
zbMATH Keywords: blow up profile
zbMATH Keywords: Cauchy-Neumann problem
MaRDI profile type: Publication
full work available at URL: j.jde.2004.10.021
OpenAlex ID: W2043336675
cites work: On the ``hot spots'' conjecture of J. Rauch
cites work: A counterexample to the ``hot spots'' conjecture
cites work: The Blow-Up Time for Solutions of Nonlinear Heat Equations with Small Diffusion
cites work: Q5547841
cites work: Blow-up time and blow-up set of the solutions for semilinear heat equations with large diffusion.
cites work: Blow-up behavior for semilinear heat equations with boundary conditions.
cites work: Location of blow-up set for a semilinear parabolic equation with large diffusion
cites work: The “hot spots” conjecture for domains with two axes of symmetry
cites work: Rearrangements and convexity of level sets in PDE
cites work: Multiple blowup of solutions for a semilinear heat equation
cites work: Life span of solutions for a semilinear parabolic problem with small diffusion
cites work: Blow up profile of a solution for a nonlinear heat equation with small diffusion
cites work: Variable instability of a constant blow up solution in a nonlinear heat equation
cites work: Stability of the blow-up profile of nonlinear heat equations from the dynamical system point of view
cites work: Solution of a nonlinear heat equation with arbitrarily given blow-up points
DOI: 10.1016/J.JDE.2004.10.021
Recommended article: Blow-up time and blow-up set of the solutions for semilinear heat equations with large diffusion.
Recommended article: Blowup rate of solutions for a semilinear heat equation with the Neumann boundary condition
Recommended article: Higher Dimensional Blow Up for Semilinear Parabolic Equations
Recommended article: Life span of solutions for a quasilinear parabolic equation with initial data having positive limit inferior at infinity
Recommended article: The blow-up problem for a semilinear parabolic equation with a potential
Recommended article: Q3822580
Recommended article: The blow-up time for higher order semilinear parabolic equations with small leading coefficients
Recommended article: Existence and large time behaviour of finite points blow-up solutions of the fast diffusion equation
Recommended article: Classification of Singularities for Blowing up Solutions in Higher Dimensions
Recommended article: Blow up profile of a solution for a nonlinear heat equation with small diffusion
description: scientific article; zbMATH DE number 2164825