Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities

Pages: 265 - 290, Volume 26, Issue 1, January 2010      doi:10.3934/dcds.2010.26.265

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Xiaojie Hou - Department of Mathematics and Statistics, University of North Carolina at Wilmington, Wilmington, NC 28403, United States (email)
Yi Li - Department of Mathematics, The University of Iowa, Iowa City, IA 52242, United States (email)
Kenneth R. Meyer - Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, United States (email)

Abstract: This paper studies the traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. The existence, uniqueness, asymptotics as well as the stability of the wave solutions are investigated. The traveling wave solutions, existed for a continuance of wave speeds, do not approach the equilibria exponentially with speed larger than the critical one. While with the critical speed, the wave solutions approach to one equilibrium exponentially fast and to the other equilibrium algebraically. This is in sharp contrast with the asymptotic behaviors of the wave solutions of the classical KPP and $m-th$ order Fisher equations. A delicate construction of super- and sub-solution shows that the wave solution with critical speed is globally asymptotically stable. A simpler alternative existence proof by LaSalle's Wazewski principle is also provided in the last section.

Keywords:  Traveling Wave, Existence, Asymptotics, Uniqueness, Heteroclinc Orbits.
Mathematics Subject Classification:  Primary: 35B35, Secondary: 35K57, 35B40, 35P15.

Received: November 2008;      Revised: July 2009;      Available Online: October 2009.