Symmetry properties of sign-changing solutions to nonlinear parabolic\n equations in unbounded domains

2021/04/09 by Juraj Földes, Földes, Juraj, Alberto Saldaña +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.2104.04555

Abstract

We study the asymptotic (in time) behavior of positive and sign-changing\nsolutions to nonlinear parabolic problems in the whole space or in the exterior\nof a ball with Dirichlet boundary conditions. We show that, under suitable\nregularity and stability assumptions, solutions are asymptotically (in time)\nfoliated Schwarz symmetric, i.e., all elements in the associated omega-limit\nset are axially symmetric with respect to a common axis passing through the\norigin and are nonincreasing in the polar angle. We also obtain symmetry\nresults for solutions of H 'enon-type problems, for equilibria (i.e. for\nsolutions of the corresponding elliptic problem), and for time periodic\nsolutions.\n

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