DISCRETE AND CONTINUOUS doi:10.3934/dcds.2009.26.265 DYNAMICAL SYSTEMS Volume 26, Number 1, January 2010 pp. 265–290 TRAVELING WAVE SOLUTIONS FOR A REACTION DIFFUSION EQUATION WITH DOUBLE DEGENERATE NONLINEARITIES Xiaojie Hou Department of Mathematics and Statistics University of North Carolina at Wilmington Wilmington, NC 28403, USA Yi Li Department of Mathematics The University of Iowa Iowa City, IA 52242, USA Department of Mathematics Xi’an Jiaotong University Xi’an, China Kenneth R. Meyer Department of Mathematical Sciences University of Cincinnati Cincinnati, OH 45221, USA (Communicated by Fanghua Lin) Abstract. This paper studies the traveling wave solutions for a reaction dif- fusion equation with double degenerate nonlinearities. The existence, unique- ness, 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 asymp- totically stable. A simpler alternative existence proof by LaSalle’s Wazewski principle is also provided in the last section. 1. Introduction. We study the asymptotic behaviors and the stability of the trav- eling wave solutions of the reaction-diffusion equation    u t = u xx + f (u), u(x, 0) = ψ(x), x ∈ R,t ∈ R + (1) with the nonlinear term f satisfying the following conditions : 2000 Mathematics Subject Classification. Primary: 35B35, Secondary: 35K57, 35B40, 35P15. Key words and phrases. Traveling Wave, Existence, Asymptotics, Uniqueness, Heteroclinc Orbits. 265