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

Abstract

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.