Journal of the European Mathematical Society

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Volume 12, Issue 2, 2010, pp. 279–312
DOI: 10.4171/JEMS/198

Published online: 2010-03-16

Convergence and sharp thresholds for propagation in nonlinear diffusion problems

Yihong Du[1] and Hiroshi Matano[2]

(1) School of Science and Technology, Armidale, Australia
(2) University of Tokyo, Japan

We study the Cauchy problem

ut = uxx + f(u) (t > 0, x ∈ ℝ1), u(0, x) = u0(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.

Keywords: Nonlinear diffusion equation, asymptotic behavior, omega limit set, Cauchy problem, Allen–Cahn, combustion, sharp threshold

Du Yihong, Matano Hiroshi: Convergence and sharp thresholds for propagation in nonlinear diffusion problems. J. Eur. Math. Soc. 12 (2010), 279-312. doi: 10.4171/JEMS/198