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

We consider the problem of the speed selection mechanism for the one-dimensional nonlinear diffusion equation $u_t=u_{\mathrm{xx}}+f\left(u\right)\mathrm{}$. 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 $2\sqrt{f^{\prime }\mathrm{}\left(0\right)\mathrm{}}\le c^{*}<2\mathrm{}\sqrt{supop\left(\frac{f\left(u\right)\mathrm{}}{u}\right)}$. The lower value $c_L=2\mathrm{}\sqrt{f^{\prime }\mathrm{}\left(0\right)\mathrm{}}$ 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.