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(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 root f'(0) less than or equal to c(*) < 2 root sup(f(u)/u). The lower value C-L = 2 root 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.