Asymptotic Speed of Propagation for Fisher-Type Degenerate Reaction-Diffusion-Convection Equations

Abstract

Abstract The paper deals with the initial value problem for the degenerate reaction-diffusion-convection equation ut + h(u)ux = (um)xx + f(u), x ∈ ℝ, t > 0, where h is continuous, m > 1, and f is of Fisher-type. By means of comparison type techniques, we prove that the equilibrium u ≡ 1 is an attractor for all solutions with a continuous, bounded, non-negative initial condition u0(x) = u(x, 0) ≢ 0. When u0 is also compactly supported and satisfies 0 ≤ u0 ≤ 1, the convergence is such that an asymptotic estimate of the interface can be obtained. The employed techniques involve the theory of travelling-wave solutions that we improve in this context. The assumptions on f and h guarantee that the threshold speed wavefront is not stationary and we show that the asymptotic speed of the interface equals this minimal speed.