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

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  • 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.

    Mathematics Subject Classification: Primary: 35K57, 35B33.


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