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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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