Qualitative properties of coexistence and semi-trivial limit profiles of nonautonomous nonlinear parabolic Dirichlet systems

TL;DR

This paper investigates the symmetry and qualitative properties of limit profiles in nonautonomous nonlinear parabolic systems with Dirichlet conditions, introducing a new boundary point lemma and analyzing both competitive and noncompetitive cases.

Contribution

It establishes conditions under which limit profiles are foliated Schwarz symmetric and introduces a novel parabolic boundary point lemma for such systems.

Findings

Limit profiles in competitive systems are foliated Schwarz symmetric under reflectional inequalities.

A new parabolic boundary point lemma is developed for analyzing symmetry.

Radial symmetry results are extended to semi-trivial profiles in noncompetitive systems.

Abstract

We study the symmetry properties of limit profiles of nonautonomous nonlinear parabolic systems with Dirichlet boundary conditions in radial bounded domains. In the case of competitive systems, we show that if the initial profiles satisfy a reflectional inequality with respect to a hyperplane, then all limit profiles are foliated Schwarz symmetric with respect to antipodal points. One of the main ingredients in the proof is a new parabolic version of Serrin's boundary point lemma. Results on radial symmetry of semi-trivial profiles are discussed also for noncompetitive systems.