Differential and Integral Equations

Interior gradient blow-up in a semilinear parabolic equation

Sigurd B. Angenent and Marek Fila

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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

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

First available in Project Euclid: 6 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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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