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Sharp interface limit of the Fisher-KPP equation

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  • We investigate the singular limit, as $\varepsilon\to 0$, of the Fisher equation $\partial_t u=\varepsilon\Delta u + \varepsilon^{-1}u(1-u)$ in the whole 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, we show that the sharp interface limit moves by a constant speed, which is the minimal speed of some related one-dimensional travelling waves. Moreover, we obtain a new estimate of the thickness of the transition layers. We also exhibit initial data "not so small" at infinity which do not allow the interface phenomena.
    Mathematics Subject Classification: Primary: 35K57, 35B25; Secondary: 92D25.


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