Journal of Dynamics and Differential Equations

, Volume 16, Issue 4, pp 1011–1060

Traveling Waves in Diffusive Random Media

Authors

  • Wenxian  Shen
    • Department of Mathematics Auburn University
Article

DOI: 10.1007/s10884-004-7832-x

Cite this article as:
Shen, W. J Dyn Diff Equat (2004) 16: 1011. doi:10.1007/s10884-004-7832-x

Abstract

The current paper is devoted to the study of traveling waves in diffusive random media, including time and/or space recurrent, almost periodic, quasiperiodic, periodic ones as special cases. It first introduces a notion of traveling waves in general random media, which is a natural extension of the classical notion of traveling waves. Roughly speaking, a solution to a diffusive random equation is a traveling wave solution if both its propagating profile and its propagating speed are random variables. Then by adopting such a point of view that traveling wave solutions are limits of certain wave-like solutions, a general existence theory of traveling waves is established. It shows that the existence of a wave-like solution implies the existence of a critical traveling wave solution, which is the traveling wave solution with minimal propagating speed in many cases. When the media is ergodic, some deterministic \hbox{properties} of average propagating profile and average propagating speed of a traveling wave solution are derived. When the media is compact, certain continuity of the propagating profile of a critical traveling wave solution is obtained. Moreover, if the media is almost periodic, then a critical traveling wave solution is almost automorphic and if the media is periodic, then so is a critical traveling wave solution. Applications of the general theory to a bistable media are discussed. The results obtained in the paper generalize many existing ones on traveling waves.

Keywords

diffusive random mediarecurrencealmost periodicityalmost automorphytraveling wave solutionwave-like solutionrandom equilibriumrandom fixed point

Copyright information

© Springer Science+Business Media, Inc. 2004