American Institute of Mathematical Sciences

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April  2010, 26(2): 767-779. doi: 10.3934/dcds.2010.26.767

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, China
2. 

Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Received  December 2008 Revised  April 2009 Published  October 2009

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 $p$.
Keywords: Gradient blowup, Nonlinear gradient source., Blowup rate.
Mathematics Subject Classification: Primary: 35K55, 35B4.
Citation: Zhengce Zhang, Bei Hu. Gradient blowup rate for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 767-779. doi: 10.3934/dcds.2010.26.767
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