## Mathematics and Statistics Faculty Publications

#### Title

Traveling Wave Solutions of the One Dimensional Diffusion-Reaction Equation

Article

#### Publication Date

1-2010

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 $\displaystyle{m}-{t}{h}$ 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.