On nonlinear parabolic equations of very fast diffusion
type: Item
instance of: scholarly article
title: On nonlinear parabolic equations of very fast diffusion
zbMATH Open document ID: 0886.35081
author: Panagiota Daskalopoulos
author: Manuel del Pino
published in: Archive for Rational Mechanics and Analysis
publication date: 1998-04-27T00:00:00Z
review text: The authors are concerned with the weak solvability of \[ \partial_t u=\nabla\cdot(u^{m-1}\nabla u)\quad\text{in }{\mathbb{R}}^n\times (0,T),\qquad u(x,0)=f(x),\quad x\in {\mathbb{R}}^n, \tag{1} \] where \(n\in\mathbb{N}\), \(T>0\), \(m<0\) and \(f\in L^1_{\text{loc}}({\mathbb{R}}^n;[0,\infty))\). \textit{J. L. Vázquez} [J. Math. Pures Appl., IX. Sér. 71, No. 6, 503-526 (1992; Zbl 0831.35082)] has established that the solvability of the initial value problem implies \(f\not\in L^1({\mathbb{R}}^n )\), and the authors showed in an earlier paper [Indiana Univ. Math. J. 43, No. 2, 703-728, (1994; Zbl 0806.35086)] that \(\displaystyle \limsup_{R\to\infty}{1\over{R^{n-2/(1-m)}}}\int_{B_R} f\geq CT^{1/(1-m)}\) is necessary for (1) to be solvable on \((0,T)\), where \(C\) is a constant depending only on \(n\) and \(m\), and \(B_R\) is the ball with radius \(R\) and center \(0\) in \({\mathbb{R}}^n\). In this paper they obtain nonexistence provided that \[ \limsup_{R\to\infty}R^{2m/(1-m)}\int_0^R\Bigl[{{ds} \over{\omega_ns^{n-1}}}\int_{B_s}f\Bigr]<C_1 T^{1/(1-m)}\tag{2} \] with \(C_1\) an explicit constant depending only of \(n\) and \(m\) and \(\omega_n\) the surface measure of the unit sphere. The main theorem of the paper is an existence result which guarantees solvability of (1) for example in the special case when \(f\) is radially symmetric and locally bounded and \(<\) in (2) is replaced by \(\geq\). Such an assertion cannot be expected for general \(f\) as an example reveals.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 1042621
zbMATH Keywords: superdiffusion
zbMATH Keywords: nonexistence
reviewed by: Georg Hetzer
MaRDI profile type: Publication
full work available at URL: s002050050033
OpenAlex ID: W1989880199
DOI: 10.1007/S002050050033
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description: scientific article; zbMATH DE number 1042621