Traveling wave in backward and forward parabolic equations from population dynamics

Lianzhang Bao; Zhengfang Zhou

doi:10.3934/dcdsb.2014.19.1507

Article Contents

2014, Volume 19, Issue 6: 1507-1522. Doi: 10.3934/dcdsb.2014.19.1507

This issue Previous Article Next Article Symmetric periodic orbits in three sub-problems of the $N$-body problem

Traveling wave in backward and forward parabolic equations from population dynamics

Lianzhang Bao^1, and
Zhengfang Zhou^2,

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

Received: September 30, 2013

Revised: January 31, 2014

Published: July 2014

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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 $u\equiv 1$.

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Mathematics Subject Classification: Primary: 35K57, 92D25; Secondary: 34B16.

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