# 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

*u*=

_{t}*u*+

_{xx}*f*(

*u*) (

*t*> 0,

*x*∈ ℝ

^{1}),

*u*(0,

*x*) =

*u*

_{0}(

*x*) (

*x*∈ ℝ

^{1}),

*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*, 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

_{λ}*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