Journal of Dynamics and Differential Equations

, Volume 14, Issue 1, pp 139–188

Asymptotic Behavior of Positive Solutions of Random and Stochastic Parabolic Equations of Fisher and Kolmogorov Types

Authors

  • Georg Hetzer
    • Department of MathematicsAuburn University, Auburn University
  • Wenxian Shen
    • Department of MathematicsAuburn University, Auburn University
  • Shu Zhu
    • School of Mathematical SciencesPeking University
Article

DOI: 10.1023/A:1012932212645

Cite this article as:
Hetzer, G., Shen, W. & Zhu, S. Journal of Dynamics and Differential Equations (2002) 14: 139. doi:10.1023/A:1012932212645

Abstract

We study the asymptotic behavior as t→∞ of positive solutions for random and stochastic parabolic equations of Fisher and Kolmogorov type. The following alternatives are established. Either (i) all positive solutions converge to one and the same trivial equilibrium, or (ii) every positive solution is neither bounded away from the trivial equilibria nor converges to them, or (iii) every positive solution is bounded away from the trivial equilibria. Moreover, for the random equation, we provide in case of alternative (iii) a fairly general condition under which every positive solution converges to uniformly positive equilibria. In the stochastic case, it is proved that there is no uniformly positive equilibrium, and under an appropriate condition, (iii) never occurs.

Copyright information

© Plenum Publishing Corporation 2002