Convergence and sharp thresholds for propagation in nonlinear diffusion problems

We study the Cauchy problem u(t) = u(xx) + f(u) (t > 0; x is an element of R-1), u(0, x) = u(0)(x) (x is an element of R-1), where f(u) is a locally Lipschitz continuous function satisfying f(0) = 0. We show that any non-negative bounded solution with compactly supported initial data converges to a stationary solution as t -> infinity. Moreover, the limit is either a constant or a symmetrically decreasing stationary solution. We also consider the special case where f is a bistable nonlinearity and the case where f is a combustion type nonlinearity. Examining the behavior of a parameter-dependent solution u(lambda), we show the existence of a sharp threshold between extinction (i.e., convergence to 0) and propagation (i.e., convergence to 1). The result holds even if f has a jumping discontinuity at u = 1.