Gradient blowup rate for a semilinear parabolic equation

Zhengce Zhang; Bei Hu

doi:10.3934/dcds.2010.26.767

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2010, Volume 26, Issue 2: 767-779. Doi: 10.3934/dcds.2010.26.767

This issue Previous Article Sc-smoothness, retractions and new models for smooth spaces Next Article A structural condition for microscopic convexity principle

Gradient blowup rate for a semilinear parabolic equation

Zhengce Zhang^1, and
Bei Hu^2,

1.
College of Science, Xi’an Jiaotong University, Xi’an, 710049
2.
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Received: November 30, 2008

Revised: March 31, 2009

Published: June 2010

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Abstract

We present a one-dimensional semilinear parabolic equation $u_t=$u _xx$ +x^m |u_x|^p, p> 0, m\geq 0$, for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We show that the spatial derivative of solutions is globally bounded in the case $p\leq m+2$ while blowup occurs at the boundary when $p>m+2$. Blowup rate is also found for some range of $p$.

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Mathematics Subject Classification: Primary: 35K55, 35B40.

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