Existence and stability of traveling waves for doubly degenerate diffusion equations

This paper is concerned with the existence and stability of traveling waves for doubly degenerate diffusion equations, where the spatial diffusion operator is of the form partial derivative(x)(|partial derivative(x)u(m)|(p-2)partial derivative(x)u(m)) with m > 0 and p > 1. It is proved that, for the slow diffusion case m(p- 1) > 1, there exists a minimum wave speed c(*), such that the problem admits smooth traveling waves when wave speed c > c(*) and semi-finite traveling waves with critical wave speed c = c* while, for the fast diffusion case 0 < m(p - 1) < 1, there is no nonnegative traveling wave solution. By the weighted energy method, we also show the L-1-stability of the traveling waves.