# American Institute of Mathematical Sciences

February  2013, 33(2): 819-835. doi: 10.3934/dcds.2013.33.819

## Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space

 1 Department of Mechanics and Mathematics, V.N.Karazin Kharkiv National University, 4, Svobody Sqr., Kharkiv, 61077, Ukraine 2 Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, P.O. Box 18, 182 08 Praha, Czech Republic

Received  August 2011 Revised  April 2012 Published  September 2012

Partial differential equations with discrete (concentrated) state-dependent delays are studied. The existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of continuously differentiable (with respect to an appropriate norm) functions is used to construct a dynamical system. This subset is an analogue of the solution manifold proposed for ordinary equations in [H.-O. Walther, The solution manifold and $C_{1}$-smoothness for differential equations with state-dependent delay, J. Differential Equations, 195(1), (2003) 46--65]. The existence of a compact global attractor is proven. As far as applications are concerned, we consider the well known Mackey-Glass-type equations with diffusion, the Lasota-Wazewska-Czyzewska model, and the delayed diffusive Nicholson's blowflies equation, all with state-dependent delays.
Citation: Alexander V. Rezounenko, Petr Zagalak. Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 819-835. doi: 10.3934/dcds.2013.33.819
##### References:
 [1] N. V. Azbelev, V. P. Maksimov and L. F. Rakhmatullina, "Introduction to the Theory of Functional Differential Equations," Moscow, Nauka, 1991. [2] A. V. Babin, and M. I. Vishik, "Attractors of Evolutionary Equations," Amsterdam, North-Holland, 1992. [3] L. Boutet de Monvel, I. D. Chueshov and A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Analysis, 34 (1998), 907-925. doi: 10.1016/S0362-546X(97)00569-5. [4] N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM. J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099. [5] I. D. Chueshov, On a certain system of equations with delay, occuring in aeroelasticity, J. Soviet Math., 58 (1992), 385-390. doi: 10.1007/BF01097291. [6] I. D. Chueshov and A. V. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations, C. R. Acad. Sci. Paris, Ser. I, 321 (1995), 607-612; ( detailedversion: Math. Physics, Analysis, Geometry, 2 (1995), 363-383). [7] I. D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," Acta, Kharkov, 1999. (in Russian). English transl. Acta, Kharkov 200). (see http://www.emis.de/monographs/Chueshov). [8] O. Diekmann, S. A. van Gils, S. Verduyn Lunel and H-O. Walther, "Delay Equations: Functional, Complex, and NonlinearAnalysis," Springer-Verlag, New York, 1995. [9] T. Faria and S. Trofimchuk, Nonmonotone travelling waves in a single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376. [10] S. A. Gourley, J. So and J. Wu, Non-locality of reaction diffusion equations induced by delay: biological modeling and nonlinear dynamics, in "D. V. Anosov, A. Skubachevskii" (Eds.), Contemporary Mathematics, Thematic Surveys, Kluwer, Plenum, Dordrecht, NewYork, 2003, 84-120; (see also: Journal of Mathematical Sciences, 124 (2004), 5119-5153). [11] J. Hadamard, "Sur les Problèmes aux Derivees partielles et Leur Signification Physique," Bull. Univ. Princeton, 13, 1902. [12] J. Hadamard, "Le Problème de Cauchy et Les èquations aux Derivees Partielles Linéaires Hyperboliques," Hermann, Paris, 1932. [13] J. K. Hale, "Theory of Functional Differential Equations," Springer, Berlin- Heidelberg- New York, 1977. [14] J. K. Hale and S. M. Verduyn Lunel, "Theory of Functional Differential Equations," Springer-Verlag, New York, 1993. [15] F. Hartung, T. Krisztin, H. O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, in "Handbookof Differential Equations: Ordinary Differential Equations, Volume 3" (eds. A. Canada, P. Drabek and A. Fonda), Elsevier B. V., 2006. [16] E. Hernandez, A. Prokopczyk and L. Ladeira, Anote on partial functional differential equations with state-dependent delay, Nonlinear Anal. R. W. A., 7 (2006), 510-519. doi: 10.1016/j.nonrwa.2005.03.014. [17] A. Lasota, Ergodic problems in biology, Dynamical systems, Warsaw, Aste'risque, Soc. Math. France, Paris II (1977), 239-250. [18] J. L. Lions and E. Magenes, "Problèmes aux Limites Non Homogénes et Applications," Dunon, Paris, 1968. [19] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Paris, 1969. [20] T. Krisztin, A local unstable manifold for differential equations with state-dependent delay, Discrete Contin.Dyn. Syst., 9 (2003), 933-1028. doi: 10.3934/dcds.2003.9.993. [21] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289. doi: 10.1126/science.267326. [22] J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal., 3 (1994), 101-162. [23] A. D. Myshkis, "Linear Differential Equations with Retarded Argument," 2nd edition, Nauka, Moscow, 1972. [24] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 1983. [25] A. V. Rezounenko, On singular limit dynamics for a class of retarded nonlinear partial differential equations, Matematicheskaya fizika, analiz, geometriya, 4 (1997), 193-211. [26] A. V. Rezounenko and J. Wu, A non-local PDE model for population dynamics with state-selective delay: local theory and global attractors, Journal of Computational and Applied Mathematics, 190 (2006), 99-113. doi: 10.1016/j.cam.2005.01.047. [27] A. V. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays, Journal of Mathematical Analysis and Applications, 326 (2007), 1031-1045. (see also detailed preprint, March 22, 2005, arXiv:math/0503470). doi: 10.1016/j.jmaa.2006.03.049. [28] A. V. Rezounenko, On a class of P.D.E.swith nonlinear distributed in space and time state-dependent delay terms, Mathematical Methods in the Applied Sciences,, 31 (2008), 1569-1585. doi: 10.1002/mma.986. [29] A. V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posednessin the space of continuous functions, Nonlinear Analysis: Theory, Methods and Applications, 70 (2009), 3978-3986. doi: 10.1016/j.na.2008.08.006. [30] A. V. Rezounenko, Non-linear partial differential equations with discrete state-dependent delays in a metric space, Nonlinear Analysis: Theory, Methods and Applications, 73 (2010), 1707-1714; (see detailed Preprint, April 15, 2009, arXiv:0904.2308). doi: 10.1016/j.na.2010.05.005. [31] R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," AMS, Mathematical Surveysand Monographs, 49 1997. [32] J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Mat. Pura ed Appl., 146 (1987), 65-96. [33] J. W. H. So, J. H. Wu and X. F. Zou, A reaction diffusion model for a single species with age structure. I. Travelling wavefronts on unbounded domains, Proc. Royal. Soc. Lond.A, 457 (2001), 1841-1853. doi: 10.1098/rspa.2001.0789. [34] J. W. H. So and Y. Yang, Dirichlet problem for the diffusive Nicholson's blowflies equation, J. Differential Equations, 150 (1998), 317-348. [35] R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Springer, Berlin-Heidelberg-New York, 1988. doi: 10.1007/978-1-4684-0313-8. [36] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Transactions of AMS, 200 (1974), 395-418. [37] H. O. Walther, Stable periodic motion of a system with state-dependent delay, Differential and Integral Equations, 15 (2002), 923-944. [38] H. O. Walther, The solution manifold and C1-smoothness for differential equations with state-dependent delay, J. Differential Equations, 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001. [39] H. O. Walther, On a model for soft landing with state-dependent delay, J. Dynamics and Differential Eqs, 19 (2007), 593-622. doi: 10.1007/s10884-006-9064-8. [40] H. O. Walther, Linearized stability for semiflows generated by a class of neutral equations, with applications to state-dependent delays, Journal of Dynamics and Differential Equations, 22 (2010), 439-462. doi: 10.1007/s10884-010-9168-z. [41] X. Wang and Z. Li, Dynamics for a type of general reaction-diffusion model, Nonlinear Analysis, 67 (2007), 2699-2711. doi: 10.1016/j.na.2006.09.034. [42] E. Winston, Uniqueness of the zero solution for differential equations with state-dependence, J. Differential Equations, 7 (1970), 395-405. doi: 10.1016/0022-0396(70)90118-X. [43] J. Wu, "Theory and Applications of Partial Functional Differential Equations," Springer-Verlag, New York, 1996. [44] S.-L. Wu, H.-Q. Zhao and S.-Y. Liu, Asymptotic stability of traveling waves for delay edreaction-diffusion equations with crossing-monostability, Z. Angew.Math. Phys., 62 (2011), 377-397. doi: 10.1007/s00033-010-0112-1. [45] K. Yosida, "Functional Analysis," Springer-Verlag, NewYork, 1965.

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##### References:
 [1] N. V. Azbelev, V. P. Maksimov and L. F. Rakhmatullina, "Introduction to the Theory of Functional Differential Equations," Moscow, Nauka, 1991. [2] A. V. Babin, and M. I. Vishik, "Attractors of Evolutionary Equations," Amsterdam, North-Holland, 1992. [3] L. Boutet de Monvel, I. D. Chueshov and A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Analysis, 34 (1998), 907-925. doi: 10.1016/S0362-546X(97)00569-5. [4] N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM. J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099. [5] I. D. Chueshov, On a certain system of equations with delay, occuring in aeroelasticity, J. Soviet Math., 58 (1992), 385-390. doi: 10.1007/BF01097291. [6] I. D. Chueshov and A. V. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations, C. R. Acad. Sci. Paris, Ser. I, 321 (1995), 607-612; ( detailedversion: Math. Physics, Analysis, Geometry, 2 (1995), 363-383). [7] I. D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," Acta, Kharkov, 1999. (in Russian). English transl. Acta, Kharkov 200). (see http://www.emis.de/monographs/Chueshov). [8] O. Diekmann, S. A. van Gils, S. Verduyn Lunel and H-O. Walther, "Delay Equations: Functional, Complex, and NonlinearAnalysis," Springer-Verlag, New York, 1995. [9] T. Faria and S. Trofimchuk, Nonmonotone travelling waves in a single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376. [10] S. A. Gourley, J. So and J. Wu, Non-locality of reaction diffusion equations induced by delay: biological modeling and nonlinear dynamics, in "D. V. Anosov, A. Skubachevskii" (Eds.), Contemporary Mathematics, Thematic Surveys, Kluwer, Plenum, Dordrecht, NewYork, 2003, 84-120; (see also: Journal of Mathematical Sciences, 124 (2004), 5119-5153). [11] J. Hadamard, "Sur les Problèmes aux Derivees partielles et Leur Signification Physique," Bull. Univ. Princeton, 13, 1902. [12] J. Hadamard, "Le Problème de Cauchy et Les èquations aux Derivees Partielles Linéaires Hyperboliques," Hermann, Paris, 1932. [13] J. K. Hale, "Theory of Functional Differential Equations," Springer, Berlin- Heidelberg- New York, 1977. [14] J. K. Hale and S. M. Verduyn Lunel, "Theory of Functional Differential Equations," Springer-Verlag, New York, 1993. [15] F. Hartung, T. Krisztin, H. O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, in "Handbookof Differential Equations: Ordinary Differential Equations, Volume 3" (eds. A. Canada, P. Drabek and A. Fonda), Elsevier B. V., 2006. [16] E. Hernandez, A. Prokopczyk and L. Ladeira, Anote on partial functional differential equations with state-dependent delay, Nonlinear Anal. R. W. A., 7 (2006), 510-519. doi: 10.1016/j.nonrwa.2005.03.014. [17] A. Lasota, Ergodic problems in biology, Dynamical systems, Warsaw, Aste'risque, Soc. Math. France, Paris II (1977), 239-250. [18] J. L. Lions and E. Magenes, "Problèmes aux Limites Non Homogénes et Applications," Dunon, Paris, 1968. [19] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Paris, 1969. [20] T. Krisztin, A local unstable manifold for differential equations with state-dependent delay, Discrete Contin.Dyn. Syst., 9 (2003), 933-1028. doi: 10.3934/dcds.2003.9.993. [21] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289. doi: 10.1126/science.267326. [22] J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal., 3 (1994), 101-162. [23] A. D. Myshkis, "Linear Differential Equations with Retarded Argument," 2nd edition, Nauka, Moscow, 1972. [24] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 1983. [25] A. V. Rezounenko, On singular limit dynamics for a class of retarded nonlinear partial differential equations, Matematicheskaya fizika, analiz, geometriya, 4 (1997), 193-211. [26] A. V. Rezounenko and J. Wu, A non-local PDE model for population dynamics with state-selective delay: local theory and global attractors, Journal of Computational and Applied Mathematics, 190 (2006), 99-113. doi: 10.1016/j.cam.2005.01.047. [27] A. V. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays, Journal of Mathematical Analysis and Applications, 326 (2007), 1031-1045. (see also detailed preprint, March 22, 2005, arXiv:math/0503470). doi: 10.1016/j.jmaa.2006.03.049. [28] A. V. Rezounenko, On a class of P.D.E.swith nonlinear distributed in space and time state-dependent delay terms, Mathematical Methods in the Applied Sciences,, 31 (2008), 1569-1585. doi: 10.1002/mma.986. [29] A. V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posednessin the space of continuous functions, Nonlinear Analysis: Theory, Methods and Applications, 70 (2009), 3978-3986. doi: 10.1016/j.na.2008.08.006. [30] A. V. Rezounenko, Non-linear partial differential equations with discrete state-dependent delays in a metric space, Nonlinear Analysis: Theory, Methods and Applications, 73 (2010), 1707-1714; (see detailed Preprint, April 15, 2009, arXiv:0904.2308). doi: 10.1016/j.na.2010.05.005. [31] R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," AMS, Mathematical Surveysand Monographs, 49 1997. [32] J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Mat. Pura ed Appl., 146 (1987), 65-96. [33] J. W. H. So, J. H. Wu and X. F. Zou, A reaction diffusion model for a single species with age structure. I. Travelling wavefronts on unbounded domains, Proc. Royal. Soc. Lond.A, 457 (2001), 1841-1853. doi: 10.1098/rspa.2001.0789. [34] J. W. H. So and Y. Yang, Dirichlet problem for the diffusive Nicholson's blowflies equation, J. Differential Equations, 150 (1998), 317-348. [35] R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Springer, Berlin-Heidelberg-New York, 1988. doi: 10.1007/978-1-4684-0313-8. [36] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Transactions of AMS, 200 (1974), 395-418. [37] H. O. Walther, Stable periodic motion of a system with state-dependent delay, Differential and Integral Equations, 15 (2002), 923-944. [38] H. O. Walther, The solution manifold and C1-smoothness for differential equations with state-dependent delay, J. Differential Equations, 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001. [39] H. O. Walther, On a model for soft landing with state-dependent delay, J. Dynamics and Differential Eqs, 19 (2007), 593-622. doi: 10.1007/s10884-006-9064-8. [40] H. O. Walther, Linearized stability for semiflows generated by a class of neutral equations, with applications to state-dependent delays, Journal of Dynamics and Differential Equations, 22 (2010), 439-462. doi: 10.1007/s10884-010-9168-z. [41] X. Wang and Z. Li, Dynamics for a type of general reaction-diffusion model, Nonlinear Analysis, 67 (2007), 2699-2711. doi: 10.1016/j.na.2006.09.034. [42] E. Winston, Uniqueness of the zero solution for differential equations with state-dependence, J. Differential Equations, 7 (1970), 395-405. doi: 10.1016/0022-0396(70)90118-X. [43] J. Wu, "Theory and Applications of Partial Functional Differential Equations," Springer-Verlag, New York, 1996. [44] S.-L. Wu, H.-Q. Zhao and S.-Y. Liu, Asymptotic stability of traveling waves for delay edreaction-diffusion equations with crossing-monostability, Z. Angew.Math. Phys., 62 (2011), 377-397. doi: 10.1007/s00033-010-0112-1. [45] K. Yosida, "Functional Analysis," Springer-Verlag, NewYork, 1965.
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