TRAVELING WAVES AND FINITE PROPAGATION IN A REACTION-DIFFUSION EQUATION

We study the existence of travelling wave solutions and the property of finite propagation for the reaction-diffusion equation ut = (um)xx + λun, (x, t) ε{lunate} R × (0, ∞) with m > 1, λ > 0, n ε{lunate} R, and u = u(x, t) ≥ 0. We show that travelling waves exist globally only if m + n = 2 and only for velocities |c| ≥ c* = 2 √λm. In the study of propagation we must take into account that solutions of the Cauchy problem can be nonunique for n < 1. Finite propagation occurs for minimal solutions if and only if m + n ≥ 2, and there exists a minimal velocity c* > 0 for m + n = 2. Maximal solutions propagate instantly to the whole space if n < 1. © 1991.