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 uₜ=uₓₓ+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√f^'(0)≤c*<2√supop(f(u)/u). The lower value cL=2√f^'(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.