JournalsjemsVol. 17, No. 10pp. 2673–2724

Spreading and vanishing in nonlinear diffusion problems with free boundaries

  • Yihong Du

    School of Science and Technology, Armidale, Australia
  • Bendong Lou

    Tongji University, Shanghai, China
Spreading and vanishing in nonlinear diffusion problems with free boundaries cover
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Abstract

We study nonlinear diffusion problems of the form ut=uxx+f(u)u_t=u_{xx}+f(u) with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special f(u)f(u) of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any f(u)f(u) which is C1C^1 and satisfies f(0)=0f(0)=0, we show that the omega limit set ω(u)\omega(u) of every bounded positive solution is determined by a stationary solution. For monostable, bistable and combustion types of nonlinearities, we obtain a rather complete description of the long-time dynamical behavior of the problem; moreover, by introducing a parameter σ\sigma in the initial data, we reveal a threshold value σ\sigma^* such that spreading (limtu=1\lim_{t \to \infty}u= 1) happens when σ>σ\sigma > \sigma^*, vanishing (limtu=0\lim_{t \to \infty}u=0) happens when σ<σ\sigma < \sigma^*, and at the threshold value σ\sigma^*, ω(u)\omega(u) is different for the three different types of nonlinearities. When spreading happens, we make use of "semi-waves" to determine the asymptotic spreading speed of the front.

Cite this article

Yihong Du, Bendong Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries. J. Eur. Math. Soc. 17 (2015), no. 10, pp. 2673–2724

DOI 10.4171/JEMS/568