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 & 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.

[2]

M. W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows,, in, 17 (1982), 267.

[3]

M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems,, J. Reine Angew. Math., 383 (1988), 1. 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., (1967).

[5]

A. Lunardi, Abstract quasilinear parabolic equations,, Math. Annalen, 267 (1984), 395. doi: 10.1007/BF01456097.

[6]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Birkhäuser Verlag, (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.

[8]

H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems,, J. Fac. Sci. Univ. Tokyo, 30 (1984), 645.

[9]

H. Matano, Strongly order-preserving local semi-dynamical systems - theory and applications,, in, 141 (1986).

[10]

H. Matano, Asymptotic behavior of solutions of semilinear heat equations on $S^1$,, in, II (1988), 139. 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. 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. 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.

[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.

[15]

P. Poláčik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations,, J. Differential Equations, 79 (1989), 89. 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. 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. 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.

show all references

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.

[2]

M. W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows,, in, 17 (1982), 267.

[3]

M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems,, J. Reine Angew. Math., 383 (1988), 1. 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., (1967).

[5]

A. Lunardi, Abstract quasilinear parabolic equations,, Math. Annalen, 267 (1984), 395. doi: 10.1007/BF01456097.

[6]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Birkhäuser Verlag, (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.

[8]

H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems,, J. Fac. Sci. Univ. Tokyo, 30 (1984), 645.

[9]

H. Matano, Strongly order-preserving local semi-dynamical systems - theory and applications,, in, 141 (1986).

[10]

H. Matano, Asymptotic behavior of solutions of semilinear heat equations on $S^1$,, in, II (1988), 139. 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. 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. 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.

[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.

[15]

P. Poláčik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations,, J. Differential Equations, 79 (1989), 89. 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. 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. 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.

[1]

M. W. Hirsch, Hal L. Smith. Asymptotically stable equilibria for monotone semiflows. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 385-398. doi: 10.3934/dcds.2006.14.385

[2]

Yurii Nesterov, Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1383-1396. doi: 10.3934/dcds.2011.31.1383

[3]

Juan A. Calzada, Rafael Obaya, Ana M. Sanz. Continuous separation for monotone skew-product semiflows: From theoretical to numerical results. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 915-944. doi: 10.3934/dcdsb.2015.20.915

[4]

Monica Lazzo, Paul G. Schmidt. Monotone local semiflows with saddle-point dynamics and applications to semilinear diffusion equations. Conference Publications, 2005, 2005 (Special) : 566-575. doi: 10.3934/proc.2005.2005.566

[5]

Marco Spadini. Branches of harmonic solutions to periodically perturbed coupled differential equations on manifolds. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 951-964. doi: 10.3934/dcds.2006.15.951

[6]

Li Ma, Chong Li, Lin Zhao. Monotone solutions to a class of elliptic and diffusion equations. Communications on Pure & Applied Analysis, 2007, 6 (1) : 237-246. doi: 10.3934/cpaa.2007.6.237

[7]

Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613

[8]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781

[9]

Dung Le. Partial regularity of solutions to a class of strongly coupled degenerate parabolic systems. Conference Publications, 2005, 2005 (Special) : 576-586. doi: 10.3934/proc.2005.2005.576

[10]

Fuqin Sun, Mingxin Wang. Non-existence of global solutions for nonlinear strongly damped hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 949-958. doi: 10.3934/dcds.2005.12.949

[11]

Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815

[12]

Sergiu Aizicovici, Simeon Reich. Anti-periodic solutions to a class of non-monotone evolution equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 35-42. doi: 10.3934/dcds.1999.5.35

[13]

Alfonso Castro, Benjamin Preskill. Existence of solutions for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 649-658. doi: 10.3934/dcds.2010.28.649

[14]

Sabri Bensid, Jesús Ildefonso Díaz. On the exact number of monotone solutions of a simplified Budyko climate model and their different stability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1033-1047. doi: 10.3934/dcdsb.2019005

[15]

Rui Li, Yuan Lou. Some monotone properties for solutions to a reaction-diffusion model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-22. doi: 10.3934/dcdsb.2019126

[16]

Mauro Patrão, Luiz A. B. San Martin. Morse decomposition of semiflows on fiber bundles. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 561-587. doi: 10.3934/dcds.2007.17.561

[17]

Y. S. Choi, Roger Lui, Yoshio Yamada. Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 719-730. doi: 10.3934/dcds.2004.10.719

[18]

Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107

[19]

Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070

[20]

Dung Le. On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3245-3265. doi: 10.3934/dcdsb.2014.19.3245

2017 Impact Factor: 1.187

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]