# Spreading and vanishing in nonlinear diffusion problems with free boundaries

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 10, page 2673-2724
- ISSN: 1435-9855

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topDu, Yihong, and Lou, Bendong. "Spreading and vanishing in nonlinear diffusion problems with free boundaries." Journal of the European Mathematical Society 017.10 (2015): 2673-2724. <http://eudml.org/doc/277414>.

@article{Du2015,

abstract = {We study nonlinear diffusion problems of the form $u_t=u_\{xx\}+f(u)$ 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 $f(u)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any $f(u)$ which is $C^1$ and satisfies $f(0)=0$, we show that the omega limit set $\omega (u)$ 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 $\sigma $ in the initial data, we reveal a threshold value $\sigma ^*$ such that spreading ($\lim _\{t \rightarrow \infty \}u= 1$) happens when $\sigma > \sigma ^*$, vanishing ($\lim _\{t \rightarrow \infty \}u=0$) happens when $\sigma < \sigma ^*$, and at the threshold value $\sigma ^*$, $\omega (u)$ 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.},

author = {Du, Yihong, Lou, Bendong},

journal = {Journal of the European Mathematical Society},

keywords = {nonlinear diffusion equation; free boundary problem; asymptotic behavior; monostable; bistable; combustion; sharp threshold; spreading speed; nonlinear diffusion equation; free boundary problem; asymptotic behavior; monostable; bistable; combustion; sharp threshold; spreading speed},

language = {eng},

number = {10},

pages = {2673-2724},

publisher = {European Mathematical Society Publishing House},

title = {Spreading and vanishing in nonlinear diffusion problems with free boundaries},

url = {http://eudml.org/doc/277414},

volume = {017},

year = {2015},

}

TY - JOUR

AU - Du, Yihong

AU - Lou, Bendong

TI - Spreading and vanishing in nonlinear diffusion problems with free boundaries

JO - Journal of the European Mathematical Society

PY - 2015

PB - European Mathematical Society Publishing House

VL - 017

IS - 10

SP - 2673

EP - 2724

AB - We study nonlinear diffusion problems of the form $u_t=u_{xx}+f(u)$ 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 $f(u)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any $f(u)$ which is $C^1$ and satisfies $f(0)=0$, we show that the omega limit set $\omega (u)$ 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 $\sigma $ in the initial data, we reveal a threshold value $\sigma ^*$ such that spreading ($\lim _{t \rightarrow \infty }u= 1$) happens when $\sigma > \sigma ^*$, vanishing ($\lim _{t \rightarrow \infty }u=0$) happens when $\sigma < \sigma ^*$, and at the threshold value $\sigma ^*$, $\omega (u)$ 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.

LA - eng

KW - nonlinear diffusion equation; free boundary problem; asymptotic behavior; monostable; bistable; combustion; sharp threshold; spreading speed; nonlinear diffusion equation; free boundary problem; asymptotic behavior; monostable; bistable; combustion; sharp threshold; spreading speed

UR - http://eudml.org/doc/277414

ER -