Asymptotic properties of positive solutions of parabolic equations and cooperative systems with Dirichlet boundary data.

Abstract

We study symmetry properties of non-negative bounded solutions of fully nonlinear parabolic equations on bounded reflectionally symmetric domains with Dirichlet boundary conditions. First we consider scalar case, and we propose sufficient conditions on the equation and domain, which guarantee asymptotic symmetry of solutions. Then we consider fully nonlinear weakly coupled systems of parabolic equations. Assuming the system is cooperative we prove the asymptotic symmetry of positive bounded solutions. To facilitate an application of the method of moving hyperplanes, we derive several estimates for linear parabolic equations and systems, such as maximum principle on small domains, Alexandrov- Krylov estimate and Harnack type estimates.