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Periodic solutions of reaction diffusion systems in a half-space domain. (English) Zbl 1146.35305
Summary: This paper is concerned with the existence and stability of periodic solutions for reaction diffusion systems with nonlinear Neumann boundary conditions in a half-space domain. The approach to the problem is by the method of upper and lower solutions and the integral representation of its associated monotone iterations. This method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iterative process in the same fashion as for parabolic initial boundary value problems. A sufficient condition for the stability of a periodic solution is given and an application is also given to a plankton allelopathic model from aquatic ecology.

##### MSC:
 35B10 Periodic solutions of PDE 35K57 Reaction-diffusion equations 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35K60 Nonlinear initial value problems for linear parabolic equations
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##### References:
 [1] Amann, H.: Periodic solutions of semilinear parabolic equations, , 1-29 (1978) · Zbl 0464.35050 [2] Brzychczy, S.: Monotone iterative methods for nonlinear parabolic and elliptic differential-functional equations, dissertations monographs N. 20, (1995) · Zbl 0843.35129 [3] Heikkila, S.; Lakshmikantham, V.: Monotone iterative techniques for discontinuous nonlinear differential equations, (1994) [4] Jiang, D.; Nieto, J. J.; Zuo, W.: On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations, J. math. Anal. appl. 289, 691-699 (2004) · Zbl 1134.34322 · doi:10.1016/j.jmaa.2003.09.020 [5] Koksal, S.; Lakshmikantham, V.: Unified approach to monotone iterative technique for semilinear parabolic problems, Dyn. contin. Discrete impuls. Syst. ser. A math. Anal. 10, 539-548 (2003) · Zbl 1024.35048 [6] Kolesov, S. Ju.: Periodic solutions of quasilinear parabolic equations of second order, Trans. Moscow math. Soc. 21, 114-146 (1970) · Zbl 0226.35040 [7] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S.: Monotone iterative techniques for nonlinear differential equations, (1985) · Zbl 0658.35003 [8] Mukhopadhyay, A.; Chattopadhyay, J.; Tapaswi, P. K.: A delay differential equations model of plankton allelopathy, Math. biosci. 149, 167-189 (1998) · Zbl 0946.92031 · doi:10.1016/S0025-5564(98)00005-4 [9] Nieto, J. J.: An abstract monotone iterative technique, Nonlinear anal. 28, 1923-1933 (1997) · Zbl 0883.47058 · doi:10.1016/S0362-546X(97)89710-6 [10] Pao, C. V.: Nonlinear parabolic and elliptic equation, (1992) · Zbl 0777.35001 [11] Pao, C. V.: Maximum principle in differential equations, J. math. Anal. appl. 217, 129-160 (1998) · Zbl 0895.35046 [12] Pao, C. V.: Periodic solutions of parabolic systems with nonlinear boundary conditions, J. math. Anal. appl. 234, 695-716 (1999) · Zbl 0932.35111 · doi:10.1006/jmaa.1999.6412 [13] Pao, C. V.: Periodic solutions of systems of parabolic equations in unbounded domains, Nonlinear anal. 40, 523-535 (2000) · Zbl 0953.35013 · doi:10.1016/S0362-546X(00)85031-2 [14] Protter, M. H.; Weinberger, H. F.: Parabolic systems in unbounded domains: I. Existence and dynamics, (1967) · Zbl 0153.13602 [15] Song, X. Y.; Chen, L. S.: Periodic solution of a delay differential equation of plankton allelopathy, Acta math. Sci. 23A, No. 1, 8-13 (2003) · Zbl 1036.34082 [16] Tian, C. R.; Xu, F.; Liu, Y.: Stable solution of a delay parabolic equation of plankton allelopathy, J. math. Res. exposition 25, 727-733 (2005) · Zbl 1090.35508 [17] Varga, R. S.: Matrix iterative analysis, (1962) [18] Vatsala, A. S.; Yang, J.: Monotone iterative technique for semilinear elliptic systems, Bound. value probabl., 93-106 (2005) · Zbl 1143.65388 · doi:10.1155/BVP.2005.93