Spreading and vanishing in nonlinear diffusion problems with free boundaries
By Yihong Du, Bendong Lou. 2013
303 citations recorded by the source corpus.
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 [8]. 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 ($\lim_{t\to\infty}u= 1$) happens when $\sigma>\sigma^*$, vanishing ($\lim_{t\to\infty}u=0$) 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.
Subfields: Modeling and Simulation, Molecular Biology, Public Health, Environmental and Occupational Health
View Paper: https://www.semanticscholar.org/paper/72c845250e82ab6ce79d3f0deb1971a696716311
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Citations in the corpus
- Free boundary problems for the diffusive competition system in higher dimension with sign-changing coefficients (2015, methodology): “For more mathematical problems with free boundary conditions, we refer the readers to [4, 6, 12, 13, 14, 31, 32] and some of the references cited therein.”
- Dynamics for a diffusive prey-predator model with different free boundaries (2015, background): “Please refer to [6]-[9], [11]-[13], [16, 21, 22, 28, 35, 37] and the references therein.”
- Free Boundary Problems for a Lotka–Volterra Competition System (2014, unknown): “For the study of free boundary problems for other type biological models, we refer to, for instance [4, 9, 10, 7, 19] and references cited therein.”
- The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries (2018, background): “These results have been extended to cases with more general f(u) in [19,27], and more accurate estimates for g(t), h(t) and u(t, x) for the spreading case have been obtained in [20].”
- On some Free Boundary Problems of the Prey-predator Model (2013, background): “Free boundary problems for the logistic type model, including the higher dimension case, heterogeneous environment case (variable coefficients, time-periodic environment) and with seasonal succession, has been studied by many authors; please refer to, for example, [6, 7, 8, 9, 11, 12, 25].”
- On a Free Boundary Problem for a Two-Species Weak Competition System (2012, background): “This setting for two competing species with a free boundary is motivated by the work of Du and Lin [11] who proposed a new approach to investigate how one species spreads and invades to a new environment (see also [9,12,14]).”
- The spreading front of invasive species in favorable habitat or unfavorable habitat (2013, background): “In [14], Gu, Lin and Lou studied how advection term (βux) affects the asymptotic spreading speeds when spreading occurs (0 β 2 √ d).”
- Regularity and Asymptotic Behavior of Nonlinear Stefan Problems (2014, unknown): “In one space dimension with bistable or combustion nonlinearities, it is shown in [10] that the limit of u is usually the stable positive steady-state except in the transition case, where the limit is a ground state (for the bistable case) or the ignition constant (for combustion nonlinearity).”