Spreading and vanishing in nonlinear diffusion problems with free boundaries

Du, 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 -