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Networks and Heterogeneous Media

2012, Volume 7, Issue 4: 673-689. doi: 10.3934/nhm.2012.7.673
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On asymptotically symmetric parabolic equations

  • 1.
    Institute for Mathematics and its Applicaitons, University of Minnesota, Minneapolis, MN 55455
  • 2.
    School of Mathematics, University of Minnesota, Minneapolis, MN 55455
  • Received: 01 January 2012 Revised: 01 June 2012

  • Primary: 35B40, 35B06; Secondary: 35K55.

  • We consider global bounded solutions of fully nonlinear parabolic equations on bounded reflectionally symmetric domains, under nonhomogeneous Dirichlet boundary condition. We assume that, as $t→∞$, the equation is asymptotically symmetric, the boundary condition is asymptotically homogeneous, and the solution is asymptotically strictly positive in the sense that all its limit profiles are strictly positive. Our main theorem states that all the limit profiles are reflectionally symmetric and decreasing on one side of the symmetry hyperplane in the direction perpendicular to the hyperplane. We also illustrate by example that, unlike for equations which are symmetric at all finite times, the result does not hold under a relaxed positivity condition requiring merely that at least one limit profile of the solution be strictly positive.

    Citation: Juraj Földes, Peter Poláčik. On asymptotically symmetric parabolic equations[J]. Networks and Heterogeneous Media, 2012, 7(4): 673-689. doi: 10.3934/nhm.2012.7.673

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  • We consider global bounded solutions of fully nonlinear parabolic equations on bounded reflectionally symmetric domains, under nonhomogeneous Dirichlet boundary condition. We assume that, as $t→∞$, the equation is asymptotically symmetric, the boundary condition is asymptotically homogeneous, and the solution is asymptotically strictly positive in the sense that all its limit profiles are strictly positive. Our main theorem states that all the limit profiles are reflectionally symmetric and decreasing on one side of the symmetry hyperplane in the direction perpendicular to the hyperplane. We also illustrate by example that, unlike for equations which are symmetric at all finite times, the result does not hold under a relaxed positivity condition requiring merely that at least one limit profile of the solution be strictly positive.


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  • This article has been cited by:

    1. J. Földes, P. Poláčik, Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains, 2015, 258, 00220396, 1859, 10.1016/j.jde.2014.11.015
    2. Juraj Földes, Alberto Saldaña, Tobias Weth, Symmetry Properties of Sign-Changing Solutions to Nonlinear Parabolic Equations in Unbounded Domains, 2021, 1040-7294, 10.1007/s10884-021-10061-x
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