Sharp asymptotics for the multidimensional KPP equation
The authors investigate asymptotics of solutions of the nonhomogeneous Kolmogorov-Petrovskij-Piskunov equation depending on a small parameter \(\varepsilon \to 0\): \[ {\partial u^\varepsilon \over \partial t}(t,x)={\varepsilon ^2\over 2}\Delta u^\varepsilon + {c(x)\over \varepsilon ^2} u^\varepsilon (1-u^\varepsilon), \quad x\in \mathbb{R}^d, \] complemented with the initial condition \(u^\varepsilon (0,x)={\mathbf 1}_A(x)\), the characteristic function of the set \(A\) which determines the initial position of the wave front. Here \(c\in C^3(\mathbb{R}^d)\) is a nonnegative function, \(c(x)\leq C(1+|x|)\). The dependence of the asymptotics on the geometry of the boundary of \(A\) is precisely described for the problem with constraints.
scientific article; zbMATH DE number 1792539
scientific article