Sharp interface limit of the Fisher-KPP equation

We investigate the singular limit, as $\text{ep}\to 0$, of the Fisher equation ${\mathrm{\partial }}_{t}u=\text{ep}\mathrm{\Delta }u+{\text{ep}}^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus, possibly, perturbations very small as $\mathrm{\parallel }x\mathrm{\parallel }\to \mathrm{\infty }$. 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. We obtain an 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.