Differential and Integral Equations

Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions

Philippe Souplet

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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.

Article information

Source
Differential Integral Equations, Volume 15, Number 2 (2002), 237-256.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060874

Mathematical Reviews number (MathSciNet)
MR1870471

Zentralblatt MATH identifier
1015.35016

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions 35K10: Second-order parabolic equations 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]

Citation

Souplet, Philippe. Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions. Differential Integral Equations 15 (2002), no. 2, 237--256. https://projecteuclid.org/euclid.die/1356060874


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