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Global existence and blowup of solutions for a parabolic equation with a gradient term


Author: Shaohua Chen
Journal: Proc. Amer. Math. Soc. 129 (2001), 975-981
MSC (1991): Primary 35K20, 35K55
DOI: https://doi.org/10.1090/S0002-9939-00-05666-5
Published electronically: December 12, 2000
MathSciNet review: 1712933
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Abstract:

The author discusses the semilinear parabolic equation $u_t=\Delta u + f(u) + g(u)\vert\nabla u\vert^2$ with $u\vert _{\partial \Omega}=0, \ u(x,0)=\phi(x)$. Under suitable assumptions on $f$ and $g$, he proves that, if $0 \leq \phi \leq \lambda \psi$ with $\lambda < 1$, then the solutions are global, while if $\phi \geq \lambda \psi$ with $\lambda > 1$, then the solutions blow up in a finite time, where $\psi$is a positive solution of $\Delta \psi+f(\psi)+g(\psi)\vert\nabla \psi\vert^2=0$, with $\psi\vert _{\partial \Omega}=0$.


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Additional Information

Shaohua Chen
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada, V5A 1S6
Email: schend@cs.sfu.ca

DOI: https://doi.org/10.1090/S0002-9939-00-05666-5
Keywords: Parabolic equation, gradient term, global existence, blowup
Received by editor(s): March 2, 1999
Published electronically: December 12, 2000
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2000 American Mathematical Society

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