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

Pages: 927 - 937, Volume 20, Issue 4, April 2008

doi:10.3934/dcds.2008.20.927       Abstract        Full Text (187.3K)       Related Articles

Jong-Shenq Guo - Department of Mathematics, National Taiwan Normal University, 88, S-4, Ting Chou Road, Taipei 11677, Taiwan (email)
Bei Hu - Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556, United States (email)

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.

Keywords:  gradient blowup, blowup rate, nonlinear gradient source.
Mathematics Subject Classification:  Primary: 35K55; Secondary: 35B35.

Received: March 2007;      Revised: September 2007;      Published: January 2008.