Similarity stabilizes blow up.
type: Item
instance of: scholarly article
title: Similarity stabilizes blow up.
zbMATH Open document ID: 1036.35106
DOI: 10.1016/S0022-0396(03)00036-6
author: Steven Schochet
published in: Journal of Differential Equations
publication date: 2003-07-30T00:00:00Z
review text: This is an extended version of the author's earlier paper [Journées Équations aux Derivées Partielles, Université de Nantes, Exp. No. 12 (1999; Zbl 1004.35062)]. The considered topic is the equation \[ \psi u_t = -L\left( | u| ^{m-1} u \right), \tag{1} \] where \(\psi(x)\) is positive and \(L\) is a self-adjoint strongly elliptic differential operator of order \(2s\). Application of the similarity transformation \[ u(t,x) = \Phi \bigl( \tau(t) \bigr) v \bigl( \tau(t), x \bigr) \qquad (\Phi > 0) \] with \[ \tau'(t) = \Phi \bigl( \tau(t) \bigr)^{m-1} \] yields the equation \[ \psi v_{\tau} = - L \left( | v| ^{m-1} v\right) - \lambda (\tau) \psi v. \tag{2} \] After defining \(\lambda\) by the condition that \(\int \psi | v| ^{m+1}\) be conserved it is proved that the solution of the rescaled equation (2) tends for \(\tau \to \infty\) to a weak solution of the equation \[ 0 = -L \left( | v| ^{m-1} v \right) - \lambda _{\infty} \psi v \] where \(\lambda_{\infty} = \lim_{\tau \to \infty} \lambda(\tau)\). This generalizes the result of \textit{C. Cortázar}, \textit{M. Del Pino} and \textit{M. Elgueta} [Indiana Univ. Math. J. 47, 541--561 (1998; Zbl 0916.35056)]. In the sequel a special case of (1) with \(L\) of order two is considered. The main theorem claims that, under appropriate assumptions, the solution to (1) blows up in finite time while the solution to the rescaled equation (2) remains bounded.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K25
zbMATH DE Number: 1957259
zbMATH Keywords: quasilinear parabolic partial differential equation
zbMATH Keywords: energy estimates
reviewed by: Hanna Marcinkowska
MaRDI profile type: Publication
cites work: On classification of blow-up patterns for a quasilinear heat equation
cites work: On the blow-up set for u_t=du^m+u^m, m>1
cites work: Symmetry in an elliptic problem and the blow—up set of
cites work: Symmetry of the blow‐up set of a porous medium type equation
cites work: On a class of thermal blow-up patterns
cites work: Some existence and nonexistence theorems for solutions of degenerate parabolic equations
cites work: Blow-up in quasilinear parabolic equations. Transl. from the Russian by Michael Grinfeld
cites work: Q3148419
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description: scientific article; zbMATH DE number 1957259