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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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