Convergence and sharp thresholds for propagation in nonlinear diffusion problems

We study the Cauchy problem u_t = uₓₓ + f(u)\ (t > 0,\ x∈ ℝ^1),\ u(0, x) = u_0(x)\ (x∈ ℝ^1), 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 → ∞ . 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_λ , 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 .