Differential and Integral Equations

Interior gradient blow-up in a semilinear parabolic equation

Sigurd B. Angenent and Marek Fila

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Abstract

We present a one-dimensional semilinear parabolic equation for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. In our example the derivative blows up in the interior of the space interval rather than at the boundary, as in earlier examples. In the case of monotone solutions we show that gradient blow-up occurs at a single point, and we study the shape of the singularity. Our argument for gradient blow-up also provides a pair of "naive viscosity sub- and super-solutions" which violate the comparison principle.

Article information

Source
Differential Integral Equations, Volume 9, Number 5 (1996), 865-877.

Dates
First available in Project Euclid: 6 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367871520

Mathematical Reviews number (MathSciNet)
MR1392084

Zentralblatt MATH identifier
0864.35052

Subjects
Primary: 35K57: Reaction-diffusion equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B35: Stability

Citation

Angenent, Sigurd B.; Fila, Marek. Interior gradient blow-up in a semilinear parabolic equation. Differential Integral Equations 9 (1996), no. 5, 865--877. https://projecteuclid.org/euclid.die/1367871520


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