The reaction-diffusion equation with Lewis function and critical Sobolev exponent
type: Item
instance of: scholarly article
title: The reaction-diffusion equation with Lewis function and critical Sobolev exponent
zbMATH Open document ID: 1015.35017
DOI: 10.1016/S0022-247X(02)00166-X
author: Zhong Tan
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-12-16T00:00:00Z
review text: Consider the equation \(a(x)u_t-\Delta u=|u|^{p-1}u\), \(x\in\Omega\), \(t>0\), complemented by the homogeneous Dirichlet boundary condition and the initial condition \(u(\cdot,0)=u_0\). Here \(\Omega\) is a smoothly bounded domain in \(\mathbb{R}^N\), \(N\geq 3\), \(a\in L^\infty(\Omega)\) is nonnegative, \(a\not\equiv 0\), and \(p=2^*-1\), where \(2^*=2N/(N-2)\) is the critical Sobolev exponent. Let \(E\) denote the corresponding energy functional and \(\Sigma=\{u\in H^1_0(\Omega)\); \(u\geq 0\), \( E(u)<S^N/N\}\), where \(S=\min\{\|\nabla u\|_2\); \(u\in H^1(\mathbb{R}^N)\), \(\|u\|_{2^*}=1\}\) and \(\|\cdot\|_q\) denotes the norm in \(L^q(\mathbb{R}^N)\). The author shows that the solution \(u\) is global and decays to zero exponentially fast if \(u_0\in\Sigma\), \(\int_\Omega|u_0|^{2^*} dx<S^N\), while it blows up in finite time if either \(u_0\not\equiv 0\), \(E(u_0)\leq 0\), or \(u_0\in\Sigma\), \(\int_\Omega|u_0|^{2^*} dx\geq S^N\). The proof of blow-up is based on the classical concavity argument of \textit{H. A. Levine} [Arch. Ration. Mech. Anal. 51, 371-386 (1973; Zbl 0278.35052)]. The condition \(E(u)<S^N/N\) excludes the most interesting case of threshold solutions lying on the borderline between global existence and blow-up. It is known that these solutions may be global but unbounded. The paper also contains some general convergence results for global solutions (Theorems 1.4 and 1.5) but these results are obviously incorrect.
reviewed by: Pavol Quittner
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 1846001
zbMATH Keywords: nonlinear parabolic equation
zbMATH Keywords: global existence
zbMATH Keywords: blow-up
zbMATH Keywords: homogeneous Dirichlet boundary condition
MaRDI profile type: Publication
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full work available at URL: s0022-247x(02)00166-x
OpenAlex ID: W2026603878
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description: scientific article; zbMATH DE number 1846001