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arXiv · 2104.04555

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

Juraj Földes·Alberto Saldaña·Tobias Weth
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Abstract

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

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Juraj Földes, Alberto Saldaña, Tobias Weth. 2021-04-09. Symmetry properties of sign-changing solutions to nonlinear parabolic equations in unbounded domains. https://arxiv.org/abs/2104.04555

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