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2002 Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions
Philippe Souplet
Differential Integral Equations 15(2): 237-256 (2002).

Abstract

We consider nonlinear parabolic equations with gradient-dependent nonlinearities, of the form $u_t-\Delta u=F(u,\nabla u)$. These equations are studied on smoothly bounded domains of ${\mathbb R}^N$, $N\geq 1$, with arbitrary (continuous) Dirichlet boundary data. Under optimal assumptions of (superquadratic) growth of $F$ with respect to $\nabla u$, we show that gradient blow-up occurs for suitably large initial data; i.e., $\nabla u$ blows up in finite time while $u$ remains uniformly bounded. Various extensions and additional results are given. We also consider some equations where the nonlinearity is nonlocal with respect to $\nabla u$, and show that gradient blow-up usually does not occur in this case.

Citation

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Philippe Souplet. "Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions." Differential Integral Equations 15 (2) 237 - 256, 2002.

Information

Published: 2002
First available in Project Euclid: 21 December 2012

zbMATH: 1015.35016
MathSciNet: MR1870471

Subjects:
Primary: 35K55
Secondary: 35B40, 35K10, 45K05

Rights: Copyright © 2002 Khayyam Publishing, Inc.

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Vol.15 • No. 2 • 2002
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