Differential and Integral Equations

Traveling waves in time dependent bistable equations

Wenxian Shen

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The current paper is to explore traveling waves in general time dependent bistable equations. In order to do so, it first introduces a notion of traveling wave solutions in general time dependent equations, which is a natural extension of classical traveling wave solutions. The basic point of view in the paper is that traveling wave solutions are certain limits of wave-like solutions. It then introduces a notion of wave-like solutions and shows in terms of certain backward-forward limits that the existence of wave-like solutions in general time dependent equations implies the existence of traveling wave solutions. It is shown that wave-like solutions exist in time dependent bistable equations and hence traveling wave solutions exist in such equations. Moreover, it is shown that traveling wave solutions in a time dependent bistable equation are stable and unique. The results obtained in the paper extend many of the results on traveling wave solutions of time independent (periodic, almost periodic) bistable equations.

Article information

Differential Integral Equations, Volume 19, Number 3 (2006), 241-278.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35K15: Initial value problems for second-order parabolic equations 74N99: None of the above, but in this section 92D25: Population dynamics (general)


Shen, Wenxian. Traveling waves in time dependent bistable equations. Differential Integral Equations 19 (2006), no. 3, 241--278. https://projecteuclid.org/euclid.die/1356050513

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