# American Institute of Mathematical Sciences

December  2012, 7(4): 881-891. doi: 10.3934/nhm.2012.7.881

## 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, Japan 2 Department of Mathematics, Josai University, 1-1, Keyakidai, Sakado, Saitama 350-0295, Japan

Received  March 2012 Revised  July 2012 Published  December 2012

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. Networks and Heterogeneous Media, 2012, 7 (4) : 881-891. doi: 10.3934/nhm.2012.7.881
##### References:
 [1] Y. Giga, N. Ishimura and Y. Kohsaka, Spiral solutions for a weakly anisotropic curvature flow equation, Adv. Math. Sci. Appl., 12 (2002), 393-408. [2] M. W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows, in "Nonlinear Partial Differential Equations" (ed. N. H. Durham), (1982), 267-285, Contemp. Math., 17, Amer. Math. Soc., Providence, R.I., (1983). [3] M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-53. doi: 10.1515/crll.1988.383.1. [4] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc., Providence, R. I., 1967. [5] A. Lunardi, Abstract quasilinear parabolic equations, Math. Annalen, 267 (1984), 395-415. doi: 10.1007/BF01456097. [6] A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6. [7] H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ., 18 (1978), 221-227. [8] H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo, 30 (1984), 645-673. [9] H. Matano, Strongly order-preserving local semi-dynamical systems - theory and applications, in "Semigroups, Theory and Applications" I, Pitman Res. Notes Math. Ser., 141, Longman Sci. Tech., Harlow, (1986). [10] H. Matano, Asymptotic behavior of solutions of semilinear heat equations on $S^1$, in "Nonlinear Diffusion Equations and Their Equilibrium States" II ( eds. W.-M. Ni, L. A. Peletier and J. Serrin), Math. Sci. Res. Inst. Publ., 13, Springer, (1988), 139-162. doi: 10.1007/978-1-4613-9608-6_8. [11] H. Matano, K. -I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit, Netw. Heterog. Media, 1 (2006), 537-568. doi: 10.3934/nhm.2006.1.537. [12] G. Namah and J.-M. Roquejoffre, Convergence to periodic fronts in a class of semilinear parabolic equations, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 521-536. doi: 10.1007/s000300050029. [13] T. Ogiwara and H. Matano, Monotonicity and convergence results in order-preserving systems in the presence of symmetry, Discrete Contin. Dynam. Systems, 5 (1999), 1-34. [14] T. Ogiwara and K.-I. Nakamura, Spiral traveling wave solutions of nonlinear diffusion equation related to a model of spiral crystal growth, Publ. Res. Inst. Math. Sci., 39 (2003), 767-783. [15] P. Poláčik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations, J. Differential Equations, 79 (1989), 89-110. doi: 10.1016/0022-0396(89)90115-0. [16] H. L. Smith and H. R. Thieme, Quasi convergence and stability for strongly order-preserving semiflows, SIAM J. Math. Anal., 21 (1990), 673-692. doi: 10.1137/0521036. [17] H. L. Smith and H. R. Thieme, Convergence for strongly order-preserving semiflows, SIAM J. Math. Anal., 22 (1991), 1081-1101. doi: 10.1137/0522070. [18] T. I. Zelenyak, Stabilisation of solution of boundary value problems for a second-order equation with one space variable, Differ. Equations, 4 (1968), 17-22.

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##### References:
 [1] Y. Giga, N. Ishimura and Y. Kohsaka, Spiral solutions for a weakly anisotropic curvature flow equation, Adv. Math. Sci. Appl., 12 (2002), 393-408. [2] M. W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows, in "Nonlinear Partial Differential Equations" (ed. N. H. Durham), (1982), 267-285, Contemp. Math., 17, Amer. Math. Soc., Providence, R.I., (1983). [3] M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-53. doi: 10.1515/crll.1988.383.1. [4] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc., Providence, R. I., 1967. [5] A. Lunardi, Abstract quasilinear parabolic equations, Math. Annalen, 267 (1984), 395-415. doi: 10.1007/BF01456097. [6] A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6. [7] H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ., 18 (1978), 221-227. [8] H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo, 30 (1984), 645-673. [9] H. Matano, Strongly order-preserving local semi-dynamical systems - theory and applications, in "Semigroups, Theory and Applications" I, Pitman Res. Notes Math. Ser., 141, Longman Sci. Tech., Harlow, (1986). [10] H. Matano, Asymptotic behavior of solutions of semilinear heat equations on $S^1$, in "Nonlinear Diffusion Equations and Their Equilibrium States" II ( eds. W.-M. Ni, L. A. Peletier and J. Serrin), Math. Sci. Res. Inst. Publ., 13, Springer, (1988), 139-162. doi: 10.1007/978-1-4613-9608-6_8. [11] H. Matano, K. -I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit, Netw. Heterog. Media, 1 (2006), 537-568. doi: 10.3934/nhm.2006.1.537. [12] G. Namah and J.-M. Roquejoffre, Convergence to periodic fronts in a class of semilinear parabolic equations, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 521-536. doi: 10.1007/s000300050029. [13] T. Ogiwara and H. Matano, Monotonicity and convergence results in order-preserving systems in the presence of symmetry, Discrete Contin. Dynam. Systems, 5 (1999), 1-34. [14] T. Ogiwara and K.-I. Nakamura, Spiral traveling wave solutions of nonlinear diffusion equation related to a model of spiral crystal growth, Publ. Res. Inst. Math. Sci., 39 (2003), 767-783. [15] P. Poláčik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations, J. Differential Equations, 79 (1989), 89-110. doi: 10.1016/0022-0396(89)90115-0. [16] H. L. Smith and H. R. Thieme, Quasi convergence and stability for strongly order-preserving semiflows, SIAM J. Math. Anal., 21 (1990), 673-692. doi: 10.1137/0521036. [17] H. L. Smith and H. R. Thieme, Convergence for strongly order-preserving semiflows, SIAM J. Math. Anal., 22 (1991), 1081-1101. doi: 10.1137/0522070. [18] T. I. Zelenyak, Stabilisation of solution of boundary value problems for a second-order equation with one space variable, Differ. Equations, 4 (1968), 17-22.
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