On the asymptotic shape of solutions to Neumann problems for non-cooperative parabolic systems

Abstract

We consider a class of nonautonomous parabolic competition-diffusion systems on bounded radial domains under Neumann boundary conditions. We show that, if the initial profiles satisfy a reflection inequality with respect to a hyperplane, then global positive solutions are asymptotically (in time) foliated Schwarz symmetric with respect to antipodal points. Additionally, a related result for (positive and sign changing solutions) of a scalar equation with Neumann boundary conditions is given. The asymptotic shape of solutions to cooperative systems is also discussed.

30 pages. Revised version

References in corpus (1)

• The shape of extremal functions for Poincaré-Sobolev-type inequalities in a ball

Cited by in corpus (1)

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