Journal of Dynamics and Differential Equations

, Volume 16, Issue 4, pp 1011–1060 | Cite as

Traveling Waves in Diffusive Random Media

  • Wenxian  Shen


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.


diffusive random media recurrence almost periodicity almost automorphy traveling wave solution wave-like solution random equilibrium random fixed point 


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Copyright information

© Springer Science+Business Media, Inc. 2004

Authors and Affiliations

  • Wenxian  Shen
    • 1
  1. 1.Department of Mathematics Auburn UniversityUSA

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