A Competition-Diffusion System Approximation to the Classical Two-Phase Stefan Problem

Japan Journal of Industrial and Applied Mathematics (Impact Factor: 0.32). 06/2001; 18(2):161-180. DOI: 10.1007/BF03168569


A new type of competition-diffusion system with a small parameter is proposed. By singular limit analysis, it is shown that
any solution of this system converges to the weak solution of the two-phase Stefan problem with reaction terms. This result
exhibits the relation between an ecological population model and water-ice solidification problems.

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    • "The study of Stefan-like problems arising in ecology over bounded domains can be traced back to the work of [16], who studied the population segregation patterns. See also [11]. In [15], the author considered a predator–prey model with the Stefan-type condition over a bounded domain. "
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    ABSTRACT: In this paper, we focus on the dynamics for a Lotka–Volterra type weak competition system with two free boundaries, where free boundaries which may intersect each other as time evolves are used to describe the spreading of two competing species, respectively. In the weak competition case, the dynamics of this model can be classified into four cases, which forms a spreading–vanishing quartering. The notion of the minimal habitat size for spreading is introduced to determine if species can always spread. Some sufficient conditions for spreading and vanishing are established. Also, when spreading occurs, some rough estimates for spreading speed and the long-time behavior of solutions are established.
    No preview · Article · Apr 2015 · Journal of Differential Equations
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    • "We assume that the free boundaries move according to one-phase Stefan condition, which is a kind of free boundary conditions widely used in the study of melting of ice [38], wound healing [9], and population dynamics [7] [12] [13]. The derivation of one-phase or two-phase Stefan conditions in population models as singular limits of competition-diffusion systems can be found in [26] [27] etc. When β = 0 (i.e., there is no advection in the environment), the qualitative properties of the problem (P ) was studied by Du and Lin [12] for logistic nonlinearity f (u) = u(1 − u). "
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    ABSTRACT: We consider Fisher-KPP equation with advection: $u_t=u_{xx}-\beta u_x+f(u)$ for $x\in (g(t),h(t))$, where $g(t)$ and $h(t)$ are two free boundaries satisfying Stefan conditions. This equation is used to describe the population dynamics in advective environments. We study the influence of the advection coefficient $-\beta$ on the long time behavior of the solutions. We find two parameters $c_0$ and $\beta^*$ with $\beta^*>c_0>0$ which play key roles in the dynamics, here $c_0$ is the minimal speed of the traveling waves of Fisher-KPP equation. More precisely, by studying a family of the initial data $\{ \sigma \phi \}_{\sigma >0}$ (where $\phi$ is some compactly supported positive function), we show that, (1) in case $\beta\in (0,c_0)$, there exists $\sigma^*\geqslant0$ such that spreading happens when $\sigma > \sigma^*$ and vanishing happens when $\sigma \in (0,\sigma^*]$; (2) in case $\beta\in (c_0,\beta^*)$, there exists $\sigma^*>0$ such that virtual spreading happens when $\sigma>\sigma^*$ (i.e., $u(t,\cdot;\sigma \phi)\to 0$ locally uniformly in $[g(t),\infty)$ and $u(t,\cdot + ct;\sigma \phi )\to 1$ locally uniformly in $\R$ for some $c>\beta -c_0$), vanishing happens when $\sigma\in (0,\sigma^*)$, and in the transition case $\sigma=\sigma^*$, $u(t, \cdot+o(t);\sigma \phi)\to V^*(\cdot-(\beta-c_0)t )$ uniformly, the latter is a traveling wave with a "big head" near the free boundary $x=(\beta-c_0)t$ and with an infinite long "tail" on the left; (3) in case $\beta = c_0$, there exists $\sigma^*>0$ such that virtual spreading happens when $\sigma > \sigma^*$ and $u(t,\cdot;\sigma \phi)\to 0$ uniformly in $[g(t),h(t)]$ when $\sigma \in (0,\sigma^*]$; (4) in case $\beta\geqslant \beta^*$, vanishing happens for any solution.
    Full-text · Article · Jan 2015 · Journal of Functional Analysis
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    • "Among others, they obtained the global existence, uniqueness, regularity and asymptotic behavior of solutions for the problem. Later [4] [5] [8] [12] studied similar strong competitive models. Recently Du and Lin [6] and Du and Lou [7] studied a free boundary problem, which is essentially the problem (1.6) in case v ≡ 0. They constructed some semi-waves to characterize the spreading of u which represents the density of a new species. "
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    ABSTRACT: We study two systems of reaction diffusion equations with monostable or bistable type of nonlinearity and with free boundaries. These systems are used as multi-species competitive model. For two-species models, we prove the existence of a traveling wave solution which consists of two semi-waves intersecting at the free boundary. For three-species models, we also prove the existence of a traveling wave solution which, however, consists of two semi-waves and one compactly supported wave in between, each intersecting with its neighbor at the free boundary.
    Full-text · Article · May 2014 · Discrete and Continuous Dynamical Systems - Series B
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