# Convergence and sharp thresholds for propagation in nonlinear diffusion problems

### Hiroshi Matano

University of Tokyo, Japan### Yihong 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