Asymptotic speed of propagation for Fisher-type degenerate reaction-diffusion-convection equations

The paper deals with the initial-value problem for the degenerate reaction-diffusion-convection equationu_t + h(u)u_x = (u^m)_xx + f(u), x Є R, t>0, m>1,with f, h continuous and f 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 u_0(x) = u(x, 0) ≠ 0. Whenu_0 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 thiscontext. 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.