Concentration phenomena in the semilinear parabolic equation
type: Item
instance of: scholarly article
title: Concentration phenomena in the semilinear parabolic equation
zbMATH Open document ID: 1014.35040
DOI: 10.1007/BF02872281
author: Zhong Tan
published in: Science in China. Series A
publication date: 2002-08-15T00:00:00Z
review text: This paper concerns asymptotic behavior of global solutions of the following semilinear parabolic equation \[ \begin{cases} u_t-\Delta u=u^p,\quad& (x,t)\in\Omega\times(0,+\infty),\\ u=0, &(x,t)\in\partial\Omega \times(0,+\infty),\\ u=\tau u_0, &(x,t)\in\Omega \times\{0\}, \end{cases} \] where \(\Omega\) is the unit ball in \(\mathbb R^N\) (\(N\geq 3\)), \(p=(N+2)/(N-2)\), \(u_0\not\equiv 0\) is nonnegative, radial, bounded function. The author proves that there exists \(\tau_1>\tau_0>0\) such that: (a) the problem possesses a unique global solution \(u(t)\) for every \(\tau\in(0,\tau_0)\), and \(u(t)\to 0\) uniformly as \(t\to\infty\); (b) the global solution \(u(t)\) for \(\tau=\tau_0\) is not uniformly bounded, and there exists a time sequence \(\{t_n\}\) so that \(\{u(t_n)\}\) and \(\{\nabla u(t_n)\}\) converge in certain sense to some concentration measures as \(t_n\to \infty\); (c) in the case \(\tau>\tau_1\) the solution \(u(t)\) blows up in finite time \(t\).
reviewed by: Ning Su
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1782038
zbMATH Keywords: critical Sobolev exponent
zbMATH Keywords: global solution
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: concentration compactness principle
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1782038