American Institute of Mathematical Sciences

August  2014, 19(6): 1507-1522. doi: 10.3934/dcdsb.2014.19.1507

Traveling wave in backward and forward parabolic equations from population dynamics

 1 College of Mathematics, Jilin University, Changchun, Jilin 130012, China 2 Department of Mathematics, Michigan State University, East Lansing, MI 48824

Received  October 2013 Revised  February 2014 Published  June 2014

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 $u\equiv 1$.
Citation: Lianzhang Bao, Zhengfang Zhou. Traveling wave in backward and forward parabolic equations from population dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1507-1522. doi: 10.3934/dcdsb.2014.19.1507
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