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Networks and Heterogeneous Media

2012, Volume 7, Issue 4: 881-891. doi: 10.3934/nhm.2012.7.881
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Periodically growing solutions in a class of strongly monotone semiflows

  • 1.
    Graduate School of Natural Science and Technology, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa 920-1192
  • 2.
    Department of Mathematics, Josai University, 1-1, Keyakidai, Sakado, Saitama 350-0295
  • Received: 01 March 2012 Revised: 01 July 2012

  • Primary: 35B40, 47H07; Secondary: 35K55.

  • We study the behavior of unbounded global orbits in a class of strongly monotone semiflows and give a criterion for the existence of orbits with periodic growth. We also prove the uniqueness and asymptotic stability of such orbits. We apply our results to a certain class of nonlinear parabolic equations including a weakly anisotropic curvature flow in a two-dimensional annulus and show the convergence of the solutions to a periodically growing solution which grows up in infinite time changing its profile time-periodically.

    Citation: Ken-Ichi Nakamura, Toshiko Ogiwara. Periodically growing solutions in a class of strongly monotone semiflows[J]. Networks and Heterogeneous Media, 2012, 7(4): 881-891. doi: 10.3934/nhm.2012.7.881

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  • We study the behavior of unbounded global orbits in a class of strongly monotone semiflows and give a criterion for the existence of orbits with periodic growth. We also prove the uniqueness and asymptotic stability of such orbits. We apply our results to a certain class of nonlinear parabolic equations including a weakly anisotropic curvature flow in a two-dimensional annulus and show the convergence of the solutions to a periodically growing solution which grows up in infinite time changing its profile time-periodically.


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