Convergence and sharp thresholds for propagation in nonlinear diffusion problems

Abstract

We study the Cauchy problem

where $f(u)$ is a locally Lipschitz continuous function satisfying $f(0)=0$. We show that any nonnegative bounded solution with compactly supported initial data converges to a stationary solution as $t\to \infty$. 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 $u=1$.