Convergence and sharp thresholds for propagation in nonlinear diffusion problems
Hiroshi Matano
University of Tokyo, JapanYihong Du
School of Science and Technology, Armidale, Australia

Abstract
We study the Cauchy problem
ut = uxx + f(u) (t > 0, x ∈ ℝ1), u(0, x) = _u_0(x) (x ∈ ℝ1),
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 uλ, 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.
Cite this article
Hiroshi Matano, Yihong Du, Convergence and sharp thresholds for propagation in nonlinear diffusion problems. J. Eur. Math. Soc. 12 (2010), no. 2, pp. 279–312
DOI 10.4171/JEMS/198