Abstract
We investigate the singular limit, as $\varepsilon\to 0$, of the Fisherequation $\partial_t u=\varepsilon\Delta u + \varepsilon^{-1}u(1-u)$ in thewhole space. We consider initial data with compact support plus,possibly, perturbations very small as $||x|| \to \infty$.By proving both generation and motion of interface properties, weshow that the sharp interface limit moves by a constant speed,which is the minimal speed of some related one-dimensionaltravelling waves. Moreover, we obtain a new estimate of thethickness of the transition layers. We also exhibit initial data'not so small' at infinity which do not allow the interfacephenomena.
Highlights
Reaction diffusion equations with logistic nonlinearity were introduced in the pioneer works of Fisher [9] or Kolmogorov, Petrovsky and Piskunov [12]
∂u (t, x) = ∆u(t, x) + u(t, x)(1 − u(t, x)), t > 0, x ∈ RN, ∂t supplemented together with some suitable initial conditions. This kind of equation was widely used in the literature to model phenomena arising in population genetics, [9, 3], or in biological invasions, [15, 14, 13] and the references therein
In order to observe such a property, we shall rescale equation (1.1) by putting uε(t, x) = u t, x . εε we focus on the singular limit problem (P ε)
Summary
Reaction diffusion equations with logistic nonlinearity were introduced in the pioneer works of Fisher [9] or Kolmogorov, Petrovsky and Piskunov [12]. The aim of this work is to focus on the ability of equation (1.1) to generate some interfaces and to propagate them Such a property is strongly related to these wave solutions. The limit solution u(t, x) will be a step function taking the value 1 on one side of the moving interface, and 0 on the other side This sharp interface, which we will denote by Γ∗t , obeys the law of motion (P ∗). The purpose of the present work is to provide a new proof of convergence for Problem (P ε) by using specific reaction-diffusion tools such as the comparison principle These technics were recently used by Hilhorst et al in [11] to consider the generation and propagation of interfaces for a degenerated Fisher equation. On the other hand we provide an O(ε| ln ε|) estimate of the thickness of the transition layers of the solutions uε
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