Periodically growing solutions  in a class of strongly monotone semiflows

Ken-Ichi Nakamura; Toshiko Ogiwara; Ken-Ichi Nakamura; Toshiko Ogiwara

doi:10.3934/nhm.2012.7.881

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

Ken-Ichi Nakamura ^{1
,},
Toshiko Ogiwara ^{2
,}

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.
- Periodically growing solutions,
- strongly monotone semiflows
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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Abstract

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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