Blow-up results for nonlinear parabolic equations on manifolds
type: Item
instance of: scholarly article
title: Blow-up results for nonlinear parabolic equations on manifolds
zbMATH Open document ID: 0954.35029
DOI: 10.1215/S0012-7094-99-09719-3
author: Qi S. Zhang
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: The author uses a unified approach to derive blow-up results for certain semilinear and quasilinear parabolic equations on manifolds. He first shows that for the equation \[ \begin{cases} \Delta u - \partial_{t}u+ V(x) u^{p} = 0, \\ u(x,0) = u_{0}(x), \quad u \geq 0, \end{cases} \] the critical exponent belongs to the blow-up case on noncompact complete manifolds. He then proves the result for the associated inhomogeneous problem, under some additional assumptions which include nonnegative Ricci curvature, as well as the existence of global positive solutions in the supercritical case. The author next generalizes blow-up results for the homogeneous porous medium equation with nonlinear source \[ \begin{cases} \Delta u^{1+\sigma} - \partial_{t}u + u^{p} = 0, \\ u(x,0) = u_{0}(x),\quad u \geq 0, \end{cases} \] to manifolds, even allowing \(\sigma\) to be somewhat negative. Notably, he does so without making any a priori assumptions on the growth of solutions. Finally, he proves that the inhomogeneous porous medium equation exhibits blow-up behavior in the subcritical case, and under additional conditions, proves the existence of global positive solutions in the supercritical case.
reviewed by: Christine Guenther
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 1425211
zbMATH Keywords: noncompact complete manifolds
zbMATH Keywords: nonnegative Ricci curvatures
zbMATH Keywords: inhomogeneous porous medium equation
zbMATH Keywords: global positive solutions
MaRDI profile type: Publication
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full work available at URL: s0012-7094-99-09719-3
OpenAlex ID: W2046812416
description: scientific article; zbMATH DE number 1425211