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March  2011, 10(2): 687-700. doi: 10.3934/cpaa.2011.10.687

Kernel sections and (almost) periodic solutions of a non-autonomous parabolic PDE with a discrete state-dependent delay

Xiang Li 1, and  Zhixiang Li 1,

1. 

Department of Mathematics and Systems Science, College of Science, National University of Defense Technology, Changsha, 410073, China, China

Received  January 2010 Revised  October 2010 Published  December 2010

In this paper, we consider the long time behavior of a non-autonomous parabolic PDE with a discrete state-dependent delay. We study the existence of compact kernel sections and unique complete trajectory of the corresponding problem. Furthermore, we obtain the (almost) periodic solution which attracts all solutions provided the time dependent terms are (almost) periodic with respect to time $t$.
Keywords: kernel section, almost periodic solution., State-dependent delay, periodic solution, complete trajectory.
Mathematics Subject Classification: 34K13, 35K9.
Citation: Xiang Li, Zhixiang Li. Kernel sections and (almost) periodic solutions of a non-autonomous parabolic PDE with a discrete state-dependent delay. Communications on Pure and Applied Analysis, 2011, 10 (2) : 687-700. doi: 10.3934/cpaa.2011.10.687
References:
[1]

W. Aiello, H. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869. doi: doi:10.1137/0152048.

[2]

Y. Cao, J. Fan and T. Gard, The effects of state-dependent time delay on a state-structured population grouwth model, Nonlinear Anal. TMA, 19 (1992), 95-105. doi: doi:10.1016/0362-546X(92)90113-S.

[3]

F. Hartung, Linearized stability in periodic functional differential equations with state-dependent delays, J. Comput. Anal. Math., 174 (2005), 201-211. doi: doi:10.1016/j.cam.2004.04.006.

[4]

R. Torrejón, Positive almost periodic solutions of a state-dependent delay nonlinear integral equation, Nonlinear Anal. TMA, 20 (1993), 1383-1416. doi: doi:10.1016/0362-546X(93)90167-Q.

[5]

H. Walther, The solution manifold and $C^1$-smoothness of solution operators for differential equations with state dependent delay, J. Differential Equations, 195 (2003), 46-65. doi: doi:10.1016/j.jde.2003.07.001.

[6]

E. Hernóndez, A. Prokopczyk and L. Ladeira, A note on partial functional differential equations with state-dependent delay, Nonlinear Anal. RWA, 7 (2006), 510-519. doi: doi:10.1016/j.nonrwa.2005.03.014.

[7]

E. Hernóndez, M. Mckibben and H. Henriquez, Existence results for partial neutral functional differential equations with state-dependent delay, Math. Comput. Modelling, 49 (2009), 1260-1267. doi: doi:10.1016/j.mcm.2008.07.011.

[8]

E. Hernóndez, M. Pierri and G. Goncalves, Existence results for an impulsive abstact partial differenrial equation with state-dependent delay, Comput. Math. Appl., 52 (2006), 411-420. doi: doi:10.1016/j.camwa.2006.03.022.

[9]

E. Hernóndez, R. Sakthivel and S. T. Aki, Existence results for impulsive evolution equation with state-dependent delay, E. J. Differential Equations, 2008 (2008), 1-11.

[10]

A. Rezounenko and J. Wu, A non-local PDE model for population dynamics with state-selective : Local theory and global attractors, J. Comput. Appl. Math., 190 (2006), 99-113. doi: doi:10.1016/j.cam.2005.01.047.

[11]

A. Rezounenko, Differential equations with discrete state dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Anal. TMA, 70 (2009), 3978-3986. doi: doi:10.1016/j.na.2008.08.006.

[12]

A. Rezounenko, Partial differential equations with disctete and distributed state-dependent delays, J. Math. Anal. Appl., 326 (2007), 1031-1045. doi: doi:10.1016/j.jmaa.2006.03.049.

[13]

A. Rezounenko, On a class of P.D.E.s with nonlinear distirbuted in space and time state-dependent delay terms, Mathematics Methods in the Applied sciences, 31 (2008), 1569-1585. doi: doi:10.1002/mma.986.

[14]

A. Rezounenko, Non-linear partial differential equations with discrete state-dependent delays in a metric space, Nonlinear Anal., 73 (2010), 1707-1714. doi: doi:10.1016/j.na.2010.05.005.

[15]

V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics," AMS Providence, Rhode Island, 2001.

[16]

J. Wu, "Theory and Applications of Partial Functional Differential Equations," Springer-verlag, New York, 1996.

[17]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Defferential Equations," Applied Mathematical Sciences, vol.44, Springer, New York, Berlin, 1983.

[18]

M. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289. doi: doi:10.1126/science.267326.

[19]

K. Gopalsamy, M. Kulenovi and G. Ladas, Oscillations and global attractivity in models of hematopoiesis, J. Dynamics and Differential Equations, 2 (1990), 117-132. doi: doi:10.1007/BF01057415.

[20]

K. Gopalsamy, S. Trofimchuk and N. Bantsur, A note on global attractivity in models of hematopoiesis, Ukrainian Mathematical J., 50 (1998), 3-12. doi: doi:10.1007/BF02514684.

[21]

X. Wang and Z. Li, Dynamics for a class of general Hematopoiesis model with periodic coefficients, Appl. Math. and Comput., 186 (2007), 460-468. doi: doi:10.1016/j.amc.2006.07.109.

[22]

X. Wang and Z. Li, Global attractivity, oscillation and Hopf bifurcation for a class of diffusive hematopoiesis models, Cent. Eur. J. Math., 5 (2007), 397-414. doi: doi:10.2478/s11533-006-0042-5.

show all references

References:
[1]

W. Aiello, H. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869. doi: doi:10.1137/0152048.

[2]

Y. Cao, J. Fan and T. Gard, The effects of state-dependent time delay on a state-structured population grouwth model, Nonlinear Anal. TMA, 19 (1992), 95-105. doi: doi:10.1016/0362-546X(92)90113-S.

[3]

F. Hartung, Linearized stability in periodic functional differential equations with state-dependent delays, J. Comput. Anal. Math., 174 (2005), 201-211. doi: doi:10.1016/j.cam.2004.04.006.

[4]

R. Torrejón, Positive almost periodic solutions of a state-dependent delay nonlinear integral equation, Nonlinear Anal. TMA, 20 (1993), 1383-1416. doi: doi:10.1016/0362-546X(93)90167-Q.

[5]

H. Walther, The solution manifold and $C^1$-smoothness of solution operators for differential equations with state dependent delay, J. Differential Equations, 195 (2003), 46-65. doi: doi:10.1016/j.jde.2003.07.001.

[6]

E. Hernóndez, A. Prokopczyk and L. Ladeira, A note on partial functional differential equations with state-dependent delay, Nonlinear Anal. RWA, 7 (2006), 510-519. doi: doi:10.1016/j.nonrwa.2005.03.014.

[7]

E. Hernóndez, M. Mckibben and H. Henriquez, Existence results for partial neutral functional differential equations with state-dependent delay, Math. Comput. Modelling, 49 (2009), 1260-1267. doi: doi:10.1016/j.mcm.2008.07.011.

[8]

E. Hernóndez, M. Pierri and G. Goncalves, Existence results for an impulsive abstact partial differenrial equation with state-dependent delay, Comput. Math. Appl., 52 (2006), 411-420. doi: doi:10.1016/j.camwa.2006.03.022.

[9]

E. Hernóndez, R. Sakthivel and S. T. Aki, Existence results for impulsive evolution equation with state-dependent delay, E. J. Differential Equations, 2008 (2008), 1-11.

[10]

A. Rezounenko and J. Wu, A non-local PDE model for population dynamics with state-selective : Local theory and global attractors, J. Comput. Appl. Math., 190 (2006), 99-113. doi: doi:10.1016/j.cam.2005.01.047.

[11]

A. Rezounenko, Differential equations with discrete state dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Anal. TMA, 70 (2009), 3978-3986. doi: doi:10.1016/j.na.2008.08.006.

[12]

A. Rezounenko, Partial differential equations with disctete and distributed state-dependent delays, J. Math. Anal. Appl., 326 (2007), 1031-1045. doi: doi:10.1016/j.jmaa.2006.03.049.

[13]

A. Rezounenko, On a class of P.D.E.s with nonlinear distirbuted in space and time state-dependent delay terms, Mathematics Methods in the Applied sciences, 31 (2008), 1569-1585. doi: doi:10.1002/mma.986.

[14]

A. Rezounenko, Non-linear partial differential equations with discrete state-dependent delays in a metric space, Nonlinear Anal., 73 (2010), 1707-1714. doi: doi:10.1016/j.na.2010.05.005.

[15]

V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics," AMS Providence, Rhode Island, 2001.

[16]

J. Wu, "Theory and Applications of Partial Functional Differential Equations," Springer-verlag, New York, 1996.

[17]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Defferential Equations," Applied Mathematical Sciences, vol.44, Springer, New York, Berlin, 1983.

[18]

M. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289. doi: doi:10.1126/science.267326.

[19]

K. Gopalsamy, M. Kulenovi and G. Ladas, Oscillations and global attractivity in models of hematopoiesis, J. Dynamics and Differential Equations, 2 (1990), 117-132. doi: doi:10.1007/BF01057415.

[20]

K. Gopalsamy, S. Trofimchuk and N. Bantsur, A note on global attractivity in models of hematopoiesis, Ukrainian Mathematical J., 50 (1998), 3-12. doi: doi:10.1007/BF02514684.

[21]

X. Wang and Z. Li, Dynamics for a class of general Hematopoiesis model with periodic coefficients, Appl. Math. and Comput., 186 (2007), 460-468. doi: doi:10.1016/j.amc.2006.07.109.

[22]

X. Wang and Z. Li, Global attractivity, oscillation and Hopf bifurcation for a class of diffusive hematopoiesis models, Cent. Eur. J. Math., 5 (2007), 397-414. doi: doi:10.2478/s11533-006-0042-5.

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