Variational estimates for the speed propagation of fronts in a nonlinear diffusive Fisher equation

We examine non-linear diffusive front propagation in the frame of the Fisher-type equation: ∂tu=∂xD(u)∂xu+u(1−u). We study the problem of a sudden jump in diffusivity motivated by models of glassy polymers. It is shown that this problem differs substantially from the problem of front propagation in the usual Fisher equation which was solved by Kolmogorov, Petrovsky, and Piskunov (KPP) in 1937. As in the Fisher, Kolmogorov, Petrovsky, Piskunov (FKPP) problem, the asymptotic dynamics of the non linear diffusive front propagation is reduced to the study of a nonlinear ordinary differential equation with adequate boundary conditions. Since this problem does not allow an exact result for the propagation speed, we use a variational approach to estimate the front speed and compare it with direct time-dependent numerical simulations showing an excellent agreement.