Gradient blowup rate for a semilinear parabolic equation

Pages: 767 - 779, Volume 26, Issue 2, February 2010

doi:10.3934/dcds.2010.26.767       Abstract        Full Text (195.0K)       Related Articles

Zhengce Zhang - College of Science, Xi’an Jiaotong University, Xi’an, 710049, China (email)
Bei Hu - Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556, United States (email)

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 $p$.

Keywords:  Gradient blowup, Blowup rate, Nonlinear gradient source.
Mathematics Subject Classification:  Primary: 35K55, 35B40.

Received: December 2008;      Revised: April 2009;      Published: October 2009.