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January  2010, 13(1): 213-227. doi: 10.3934/dcdsb.2010.13.213

Critical exponents and traveling wavefronts of a degenerate-singular parabolic equation in non-divergence form

1. 

Department of Mathematics, Jilin University, Changchun 130012, China, China

Received  September 2008 Revised  August 2009 Published  October 2009

We discuss the possible existence of uncountable smooth traveling wavefronts of a degenerate and singular parabolic equation in non-divergence form

$\frac{\partial u}{\partial t} =u^m $div$(|\nabla u|^{p-2}\nabla u)+u^qf(u),$

where $f(s)$ is a positive source taking logistic type as an example. A very interesting phenomenon is the presence of critical values $m_c$ and $q_c$ of the exponent $m$ and $q$. Precisely speaking, only for the case $m$<$m_c$ with $q\ge q_c$ can the family of smooth traveling wavefronts have minimal wave speed. We also discuss the regularity of smooth traveling wavefronts.

Citation: Jingxue Yin, Chunhua Jin. Critical exponents and traveling wavefronts of a degenerate-singular parabolic equation in non-divergence form. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 213-227. doi: 10.3934/dcdsb.2010.13.213
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