Traveling wave in backward and forward parabolic equations from population dynamics

Abstract

This work is concerned with the properties of the traveling wave of the backward and forward parabolic equation \begin{equation*} u_t= [ D(u)u_x]_x + g(u),\quad t\geq 0, x\in \mathbb{R}, \end{equation*} where $D(u)$ changes its sign once, from negative to positive value, in the interval $u\in [0,1]$ and $g(u)$ is a mono-stable nonlinear reaction term. The existence of infinitely many traveling wave solutions is proven. These traveling waves are parameterized by their wave speed and monotonically connect the stationary states $u\equiv0$ and $u\equiv 1$.