Communications on Pure and Applied Analysis (CPAA)

Gradient blowup solutions of a semilinear parabolic equation with exponential source

Pages: 269 - 280, Volume 12, Issue 1, January 2013      doi:10.3934/cpaa.2013.12.269

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Zhengce Zhang - College of Science, Xi’an Jiaotong University, Xi’an, 710049, China (email)
Yanyan Li - School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China (email)

Abstract: In this paper, we consider the N-dimensional semilinear parabolic equation $ u_t=\Delta u+e^{|\nabla u|}$, for which the spatial derivative of solutions becomes unbounded in finite (or infinite) time while the solutions themselves remain bounded. We establish estimates of blowup rate as well as lower and upper bounds for the radial solutions. We prove that in this case the blowup rate does not match the one obtained by the rescaling method.

Keywords:  Gradient blowup, rate estimate, exponential source.
Mathematics Subject Classification:  Primary: 35K05, 35K35; Secondary: 35B40.

Received: June 2011;      Revised: October 2011;      Available Online: September 2012.