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Existence and stability of periodic solutions for parabolic systems with time delays. (English) Zbl 1130.35075
Summary: This paper is concerned with the existence and stability of time-periodic solutions for a class of coupled parabolic equations with time delay. Time delays may appear in the nonlinear reaction functions. The existence of time-periodic solutions is proved for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement. An approach to solve the problem is based on the method of upper and lower solution and Schauder fixed point theorem. Some other methods for proving the stability of the periodic solution are also given. The results for the general system can be applied to standard parabolic equations without time delay and corresponding ordinary differential systems. Finally, a model arising from chemistry is used to illustrate the obtained results.
MSC:
35K50Systems of parabolic equations, boundary value problems (MSC2000)
35B10Periodic solutions of PDE
References:
[1]Tineo, A.: Asymptotic behavior of solutions of a periodic reaction – diffusion system of a competitor – competitor – mutualist model, J. differential equations 108, 326-341 (1994) · Zbl 0806.35095 · doi:10.1006/jdeq.1994.1037
[2]Amann, H.: Periodic solutions of semilinear parabolic equations, , 1-29 (1978) · Zbl 0464.35050
[3]Bange, D. W.: Periodic solutions of a quasilinear parabolic differential equation, J. differential equations 17, 61-72 (1975) · Zbl 0291.35051 · doi:10.1016/0022-0396(75)90034-0
[4]Brown, K. J.; Hess, P.: Positive periodic solutions of predator – prey reaction – diffusion systems, Nonlinear anal. 16, 1147-1158 (1991) · Zbl 0743.35030 · doi:10.1016/0362-546X(91)90202-C
[5]Du, Y.: Positive periodic solutions of a competitor – competitor – mutualist model, Differential integral equations 19, 1043-1066 (1996) · Zbl 0858.35057
[6]Feng, W.; Lu, X.: Asymptotic periodicity in logistic equations with discrete delays, Nonlinear anal. 26, 171-178 (1996) · Zbl 0842.35129 · doi:10.1016/0362-546X(94)00271-I
[7]Fife, P.: Solutions of parabolic boundary problems existing for all times, Arch. ration. Mech. anal. 16, 155-186 (1964) · Zbl 0173.38204 · doi:10.1007/BF00250642
[8]Fu, S.; Ma, R.: Existence of a global coexistence state for periodic competition diffusion systems, Nonlinear anal. 28, 1265-1271 (1997) · Zbl 0871.35051 · doi:10.1016/S0362-546X(97)82873-8
[9]Hess, P.: Periodic-parabolic boundary value problems and positivity, Pitman res. Notes math. 247 (1991) · Zbl 0731.35050
[10]Kolesov, J. S.: Periodic solutions of quasilinear parabolic equations of second order, Trans. Moscow math. Soc. 21, 114-146 (1970) · Zbl 0226.35040
[11]Leung, A. W.: Systems of nonlinear partial differential equations, (1989)
[12]Leung, A. W.; Ortega, L. A.: Existence and monotone scheme for time-periodic nonquasimonotone reaction – diffusion systems: application to autocatalytic chemistry, J. math. Anal. appl. 221, 712-733 (1998) · Zbl 0914.35145 · doi:10.1006/jmaa.1998.5943
[13]Lieberman, G. M.: Time-periodic solutions of quasilinear parabolic equations, J. math. Anal. appl. 264, 617-638 (2001)
[14]Liu, B. P.; Pao, C. V.: Periodic solutions of coupled semilinear parabolic boundary value problems, Nonlinear anal. 6, 237-252 (1982) · Zbl 0499.35012 · doi:10.1016/0362-546X(82)90092-X
[15]Lu, X.; Feng, W.: Periodic solution and oscillation in a competition model with diffusion and distributed delay effect, Nonlinear anal. 27, 699-709 (1996) · Zbl 0862.35134 · doi:10.1016/0362-546X(95)00067-6
[16]Nkashama, M. N.: Semilinear periodic – parabolic equations with nonlinear boundary conditions, J. differential equations 130, 377-405 (1996) · Zbl 0861.35048 · doi:10.1006/jdeq.1996.0150
[17]Shair, A.; Lazer, A. C.: Asymptotic behaviour of solutions of periodic competition diffusion system, Nonlinear anal. 13, 263-284 (1989) · Zbl 0686.35060 · doi:10.1016/0362-546X(89)90054-0
[18]Tineo, A.: Existence of global coexistence for periodic competition diffusion systems, Nonlinear anal. 19, 335-344 (1992) · Zbl 0779.35058 · doi:10.1016/0362-546X(92)90178-H
[19]Tineo, A.; Rivero, J.: Permanence and asymptotic stability for competition Lotka – Volterra systems with diffusion, Nonlinear anal. 4, 625-637 (2003) · Zbl 1088.35028 · doi:10.1016/S1468-1218(02)00081-0
[20]Tsai, L. Y.: Periodic solutions of nonlinear partial differential equations, Bull. inst. Math. acad. Sinica 5, 219-247 (1977) · Zbl 0375.35031
[21]Wang, Y.: Convergence to periodic solutions in periodic quasimonotone reaction diffusion systems, J. math. Anal. appl. 268, 25-40 (2002) · Zbl 1042.35030 · doi:10.1006/jmaa.2001.7777
[22]Wu, J.: Theory and applications of partial functional differential equations, (1996)
[23]Zhao, X. Q.: Global asymptotic behavior in a periodic competitor – competitor – mutualist system, Nonlinear anal. 29, 551-568 (1997) · Zbl 0876.35058 · doi:10.1016/S0362-546X(96)00056-9
[24]Zhao, X. Q.; Hutson, V.: Permanence in Kolmogorov periodic predator – prey models with diffusion, Nonlinear anal. 23, 651-668 (1994) · Zbl 0823.92031 · doi:10.1016/0362-546X(94)90244-5
[25]Zheng, S.: A reaction – diffusion system of a competitor – competitor – mutualist model, J. math. Anal. appl. 124, 254-280 (1987) · Zbl 0658.35053 · doi:10.1016/0022-247X(87)90038-2
[26]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
[27]Pao, C. V.: Periodic solutions of parabolic systems with time delays, J. math. Anal. appl. 251, 251-263 (2000) · Zbl 0967.35061 · doi:10.1006/jmaa.2000.7045
[28]Zhou, L.; Fu, Y. P.: Periodic quasimonotone global attractor of nonlinear parabolic systems with discrete delays, J. math. Anal. appl. 250, 139-161 (2000)
[29]Liu, Y. D.; Li, Z. Y.; Ye, Q. X.: The existence, uniqueness and stability of positive periodic solution for periodic reaction – diffusion system, Acta math. Appl. sin. 1, 1-13 (2001) · Zbl 1158.35009 · doi:10.1007/BF02669678
[30]He, M. X.: On periodic and almost periodic solutions of reaction diffusion system with time lag, Acta math. Sinica (Chin. Ser.) 32, 91-97 (1989) · Zbl 0673.35050
[31]Pao, C. V.: Numerical methods for time-periodic solutions of nonlinear parabolic boundary value problems, SIAM J. Numer. anal. 39, 647-667 (2001) · Zbl 1004.65092 · doi:10.1137/S0036142999361396
[32]Leung, A. W.; Ortega, L. A.: Existence and monotone scheme for time-periodic nonquasi-monotone reaction – diffusion systems: application to autocatalytic chemistry, J. math. Anal. appl. 221, 712-733 (1998) · Zbl 0914.35145 · doi:10.1006/jmaa.1998.5943
[33]Tyson, J.: Belousov – zhabotinskii reaction, (1976)
[34]Murray, J. D.: On travelliny wave solution in a model for the Belousov – zhabotinskii reaction, J. theoret. Biol. 56, 329-353 (1976)