Sharp asymptotics for the multidimensional KPP equation

In this article sharp asymptotics for the solution of nonhomogeneous Kolmogorov, Petrovskii and Pisciunov equation depending on a small parameter are considered when the initial condition is the characteristic function of a set . We show how to extend the Ben Arous and Rouault's result that dealt with d=1 and the initial condition as the characteristic function of A={x[less-than-or-equals, slant]0}. The dependance of the asymptotics on the geometry of the boundary of A is precisely described for the problem with constraints.