TY - JOUR

T1 - Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains

AU - Földes, J.

AU - Poláčik, P.

N1 - Publisher Copyright:
© 2014 Elsevier Inc.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015/3/15

Y1 - 2015/3/15

N2 - We consider the Dirichlet problem for a class of semilinear parabolic equations on a bounded domain which is reflectionally symmetric about a hyperplane H. The equations consist of a symmetric time-autonomous part and a nonsymmetric perturbation which decays to zero as time approaches infinity. In our first theorem, we prove the asymptotic symmetry of each bounded positive solution of this asymptotically symmetric problem. The novelty of this result is that the solutions considered are not assumed uniformly positive, which prevents one from applying common techniques based on the method of moving hyperplanes. In our second main theorem, we classify the positive entire solutions of the unperturbed time-autonomous problems. In particular, we characterize all entire solutions, which are not symmetrically decreasing in the direction orthogonal to H, as connecting orbits from an equilibrium with a nontrivial nodal set to another invariant set.

AB - We consider the Dirichlet problem for a class of semilinear parabolic equations on a bounded domain which is reflectionally symmetric about a hyperplane H. The equations consist of a symmetric time-autonomous part and a nonsymmetric perturbation which decays to zero as time approaches infinity. In our first theorem, we prove the asymptotic symmetry of each bounded positive solution of this asymptotically symmetric problem. The novelty of this result is that the solutions considered are not assumed uniformly positive, which prevents one from applying common techniques based on the method of moving hyperplanes. In our second main theorem, we classify the positive entire solutions of the unperturbed time-autonomous problems. In particular, we characterize all entire solutions, which are not symmetrically decreasing in the direction orthogonal to H, as connecting orbits from an equilibrium with a nontrivial nodal set to another invariant set.

KW - Asymptotic symmetry

KW - Classification of entire solutions

KW - Equilibria with a nontrivial nodal set

KW - Morse decomposition

KW - Semilinear parabolic equations

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U2 - 10.1016/j.jde.2014.11.015

DO - 10.1016/j.jde.2014.11.015

M3 - Article

AN - SCOPUS:84921541853

VL - 258

SP - 1859

EP - 1888

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 6

ER -