Blowup rate estimates for the heat equation with a nonlinear gradient source term

Jong-Shenq Guo; Bei Hu

doi:10.3934/dcds.2008.20.927

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2008, Volume 20, Issue 4: 927-937. Doi: 10.3934/dcds.2008.20.927

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Blowup rate estimates for the heat equation with a nonlinear gradient source term

Jong-Shenq Guo^1, and
Bei Hu^2,

1.
Department of Mathematics, National Taiwan Normal University, 88, S-4, Ting Chou Road, Taipei 11677
2.
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Received: February 28, 2007

Revised: August 31, 2007

Published: September 2008

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Abstract

The gradient blowup rate of the equation $u_t = \Delta u + |\nabla u|^p$, where $p>2$, is studied. It is shown that the blowup rate will never match that of the self-similar variables. In the one space dimensional case when assumptions are made on the initial data so that the solution is monotonically increasing in time, the exact blowup rate is found.

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Mathematics Subject Classification: Primary: 35K55; Secondary: 35B35.

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