An example of blowup produced by equal diffusions
type: Item
instance of: scholarly article
title: An example of blowup produced by equal diffusions
zbMATH Open document ID: 0926.35051
published in: Journal of Differential Equations
publication date: 1999-11-16T00:00:00Z
review text: The paper proposes a very elegant example of blowing up solutions of a parabolic system with bounded initial data, produced by equal diffusion constants. Precisely, the author considers the system \[ {{\partial u}\over {\partial t}}=\varepsilon {{\partial^2 u}\over {\partial x^2}}+uv(u-v)(u+1)-\delta u,\quad x\in(-1,1), \] \[ {{\partial v}\over {\partial t}}=\varepsilon {{\partial^2 v}\over {\partial x^2}}+vu(v-u)(v+1)-\delta v,\quad x\in(-1,1),\qquad {{\partial u}\over {\partial x}}= {{\partial v}\over {\partial x}}=0, \quad x=\pm 1. \] It is proved that when \(\varepsilon=0\) and \(\delta>0,\) the origin is a global attractor of the positive solutions to the above system, but when \(\varepsilon\) and \(\delta\) are both positive, there is a positive solution with bounded initial data which blows up before any prescribed positive time \(T.\)
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 1288037
zbMATH Keywords: blow-up of solutions to parabolic system
zbMATH Keywords: bounded initial data
author: Hans F. Weinberger
reviewed by: Dian K. Palagachev
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3546
OpenAlex ID: W1979693546
cites work: Q3485265
cites work: Q5544622
cites work: Diffusion terms in systems of reaction diffusion equations can lead to blow up
cites work: The Role of Critical Exponents in Blowup Theorems
cites work: Diffusion-induced blowup in a nonlinear parabolic system
cites work: The chemical basis of morphogenesis
cites work: Q4071618
DOI: 10.1006/JDEQ.1998.3546
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description: scientific article; zbMATH DE number 1288037