zbMATH — the first resource for mathematics

A PDE approach to geometric optics for certain semilinear parabolic equations. (English) Zbl 0692.35014
The authors study the asymptotic behavior for large \(| x|\) and t of solutions of a class of semilinear parabolic equations in \({\mathbb{R}}^ n\). A model problem in \({\mathbb{R}}^ 1\) is the Kolmogorov-Petrovskij- Piskunov equation \[ u_ t=(D/2)u_{xx}+u(1-u);\quad u|_{t=0}=\chi_{(-\infty,0]}, \] where \(\chi_{(-\infty,0]}\) is the indicator function of (-\(\infty,0]\). The authors want to study the limit u of the sequence \(u^{\epsilon}(x,t)=u(x/\epsilon,t/\epsilon).\)
From the properties of the function u(1-u) it is reasonable to expect that u is the indicator function of a suitable measurable set G. This is proved under general assumptions and it is shown that G is the positivity set of a certain action function I(x,t), which is also shown to be the unique solution of a variational inequality involving a Hamilton-Jacobi type PDE. The results of this paper are extensions of results of M. Freidlin, who uses probabilistic techniques to derive his results.
Reviewer: H.-D.Alber

35B40 Asymptotic behavior of solutions to PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
78A25 Electromagnetic theory (general)
Full Text: DOI