Validity of the Linear Speed Selection Mechanism for Fronts of the Nonlinear Diffusion Equation

Abstract

We consider the problem of the speed selection mechanism for the one-dimensional nonlinear diffusion equation ${u}_{t}={u}_{\mathrm{xx}}+f(u)$. It has been rigorously shown by Aronson and Weinberger that for a wide class of functions $f$, sufficiently localized initial conditions evolve in time into a monotonic front which propagates with speed ${c}^{*}$ such that $2\sqrt{{f}^{\ensuremath{'}}(0)}\ensuremath{\le}{c}^{*}<2\sqrt{supop(\frac{f(u)}{u})}$. The lower value ${c}_{L}=2\sqrt{{f}^{\ensuremath{'}}(0)}$ is that predicted by the linear marginal stability speed selection mechanism. We derive a new lower bound on the speed of the selected front, this bound depends on $f$ and thus enables us to assess the extent to which the linear marginal selection mechanism is valid.