# 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 & Continuous Dynamical Systems - A, 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, (1991).   Google Scholar [2] A. V. Babin, and M. I. Vishik, "Attractors of Evolutionary Equations,", Amsterdam, (1992).   Google Scholar [3] L. Boutet de Monvel, I. D. Chueshov and A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations,, Nonlinear Analysis, 34 (1998), 907.  doi: 10.1016/S0362-546X(97)00569-5.  Google Scholar [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.  doi: 10.1137/0150099.  Google Scholar [5] I. D. Chueshov, On a certain system of equations with delay, occuring in aeroelasticity,, J. Soviet Math., 58 (1992), 385.  doi: 10.1007/BF01097291.  Google Scholar [6] I. D. Chueshov and A. V. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations,, C. R. Acad. Sci. Paris, 321 (1995), 607.   Google Scholar [7] I. D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems,", Acta, (1999).   Google Scholar [8] O. Diekmann, S. A. van Gils, S. Verduyn Lunel and H-O. Walther, "Delay Equations: Functional, Complex, and NonlinearAnalysis,", Springer-Verlag, (1995).   Google Scholar [9] T. Faria and S. Trofimchuk, Nonmonotone travelling waves in a single species reaction-diffusion equation with delay,, J. Differential Equations, 228 (2006), 357.   Google Scholar [10] S. A. Gourley, J. So and J. Wu, Non-locality of reaction diffusion equations induced by delay: biological modeling and nonlinear dynamics,, in, 124 (2004), 84.   Google Scholar [11] J. Hadamard, "Sur les Problèmes aux Derivees partielles et Leur Signification Physique,", Bull. Univ. Princeton, 13 (1902).   Google Scholar [12] J. Hadamard, "Le Problème de Cauchy et Les èquations aux Derivees Partielles Linéaires Hyperboliques,", Hermann, (1932).   Google Scholar [13] J. K. Hale, "Theory of Functional Differential Equations,", Springer, (1977).   Google Scholar [14] J. K. Hale and S. M. Verduyn Lunel, "Theory of Functional Differential Equations,", Springer-Verlag, (1993).   Google Scholar [15] F. Hartung, T. Krisztin, H. O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications,, in, (2006).   Google Scholar [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.  doi: 10.1016/j.nonrwa.2005.03.014.  Google Scholar [17] A. Lasota, Ergodic problems in biology,, Dynamical systems, II (1977), 239.   Google Scholar [18] J. L. Lions and E. Magenes, "Problèmes aux Limites Non Homogénes et Applications,", Dunon, (1968).   Google Scholar [19] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod, (1969).   Google Scholar [20] T. Krisztin, A local unstable manifold for differential equations with state-dependent delay,, Discrete Contin.Dyn. Syst., 9 (2003), 933.  doi: 10.3934/dcds.2003.9.993.  Google Scholar [21] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system,, Science, 197 (1977), 287.  doi: 10.1126/science.267326.  Google Scholar [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.   Google Scholar [23] A. D. Myshkis, "Linear Differential Equations with Retarded Argument,", 2nd edition, (1972).   Google Scholar [24] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer-Verlag, (1983).   Google Scholar [25] A. V. Rezounenko, On singular limit dynamics for a class of retarded nonlinear partial differential equations,, Matematicheskaya fizika, 4 (1997), 193.   Google Scholar [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.  doi: 10.1016/j.cam.2005.01.047.  Google Scholar [27] A. V. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays,, Journal of Mathematical Analysis and Applications, 326 (2007), 1031.  doi: 10.1016/j.jmaa.2006.03.049.  Google Scholar [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.  doi: 10.1002/mma.986.  Google Scholar [29] A. V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posednessin the space of continuous functions,, Nonlinear Analysis: Theory, 70 (2009), 3978.  doi: 10.1016/j.na.2008.08.006.  Google Scholar [30] A. V. Rezounenko, Non-linear partial differential equations with discrete state-dependent delays in a metric space,, Nonlinear Analysis: Theory, 73 (2010), 1707.  doi: 10.1016/j.na.2010.05.005.  Google Scholar [31] R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,", AMS, 49 (1997).   Google Scholar [32] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Annali di Mat. Pura ed Appl., 146 (1987), 65.   Google Scholar [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.  doi: 10.1098/rspa.2001.0789.  Google Scholar [34] J. W. H. So and Y. Yang, Dirichlet problem for the diffusive Nicholson's blowflies equation,, J. Differential Equations, 150 (1998), 317.   Google Scholar [35] R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar [36] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations,, Transactions of AMS, 200 (1974), 395.   Google Scholar [37] H. O. Walther, Stable periodic motion of a system with state-dependent delay,, Differential and Integral Equations, 15 (2002), 923.   Google Scholar [38] H. O. Walther, The solution manifold and C1-smoothness for differential equations with state-dependent delay,, J. Differential Equations, 195 (2003), 46.  doi: 10.1016/j.jde.2003.07.001.  Google Scholar [39] H. O. Walther, On a model for soft landing with state-dependent delay,, J. Dynamics and Differential Eqs, 19 (2007), 593.  doi: 10.1007/s10884-006-9064-8.  Google Scholar [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.  doi: 10.1007/s10884-010-9168-z.  Google Scholar [41] X. Wang and Z. Li, Dynamics for a type of general reaction-diffusion model,, Nonlinear Analysis, 67 (2007), 2699.  doi: 10.1016/j.na.2006.09.034.  Google Scholar [42] E. Winston, Uniqueness of the zero solution for differential equations with state-dependence,, J. Differential Equations, 7 (1970), 395.  doi: 10.1016/0022-0396(70)90118-X.  Google Scholar [43] J. Wu, "Theory and Applications of Partial Functional Differential Equations,", Springer-Verlag, (1996).   Google Scholar [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.  doi: 10.1007/s00033-010-0112-1.  Google Scholar [45] K. Yosida, "Functional Analysis,", Springer-Verlag, (1965).   Google Scholar

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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, (1991).   Google Scholar [2] A. V. Babin, and M. I. Vishik, "Attractors of Evolutionary Equations,", Amsterdam, (1992).   Google Scholar [3] L. Boutet de Monvel, I. D. Chueshov and A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations,, Nonlinear Analysis, 34 (1998), 907.  doi: 10.1016/S0362-546X(97)00569-5.  Google Scholar [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.  doi: 10.1137/0150099.  Google Scholar [5] I. D. Chueshov, On a certain system of equations with delay, occuring in aeroelasticity,, J. Soviet Math., 58 (1992), 385.  doi: 10.1007/BF01097291.  Google Scholar [6] I. D. Chueshov and A. V. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations,, C. R. Acad. Sci. Paris, 321 (1995), 607.   Google Scholar [7] I. D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems,", Acta, (1999).   Google Scholar [8] O. Diekmann, S. A. van Gils, S. Verduyn Lunel and H-O. Walther, "Delay Equations: Functional, Complex, and NonlinearAnalysis,", Springer-Verlag, (1995).   Google Scholar [9] T. Faria and S. Trofimchuk, Nonmonotone travelling waves in a single species reaction-diffusion equation with delay,, J. Differential Equations, 228 (2006), 357.   Google Scholar [10] S. A. Gourley, J. So and J. Wu, Non-locality of reaction diffusion equations induced by delay: biological modeling and nonlinear dynamics,, in, 124 (2004), 84.   Google Scholar [11] J. Hadamard, "Sur les Problèmes aux Derivees partielles et Leur Signification Physique,", Bull. Univ. Princeton, 13 (1902).   Google Scholar [12] J. Hadamard, "Le Problème de Cauchy et Les èquations aux Derivees Partielles Linéaires Hyperboliques,", Hermann, (1932).   Google Scholar [13] J. K. Hale, "Theory of Functional Differential Equations,", Springer, (1977).   Google Scholar [14] J. K. Hale and S. M. Verduyn Lunel, "Theory of Functional Differential Equations,", Springer-Verlag, (1993).   Google Scholar [15] F. Hartung, T. Krisztin, H. O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications,, in, (2006).   Google Scholar [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.  doi: 10.1016/j.nonrwa.2005.03.014.  Google Scholar [17] A. Lasota, Ergodic problems in biology,, Dynamical systems, II (1977), 239.   Google Scholar [18] J. L. Lions and E. Magenes, "Problèmes aux Limites Non Homogénes et Applications,", Dunon, (1968).   Google Scholar [19] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod, (1969).   Google Scholar [20] T. Krisztin, A local unstable manifold for differential equations with state-dependent delay,, Discrete Contin.Dyn. Syst., 9 (2003), 933.  doi: 10.3934/dcds.2003.9.993.  Google Scholar [21] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system,, Science, 197 (1977), 287.  doi: 10.1126/science.267326.  Google Scholar [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.   Google Scholar [23] A. D. Myshkis, "Linear Differential Equations with Retarded Argument,", 2nd edition, (1972).   Google Scholar [24] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer-Verlag, (1983).   Google Scholar [25] A. V. Rezounenko, On singular limit dynamics for a class of retarded nonlinear partial differential equations,, Matematicheskaya fizika, 4 (1997), 193.   Google Scholar [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.  doi: 10.1016/j.cam.2005.01.047.  Google Scholar [27] A. V. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays,, Journal of Mathematical Analysis and Applications, 326 (2007), 1031.  doi: 10.1016/j.jmaa.2006.03.049.  Google Scholar [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.  doi: 10.1002/mma.986.  Google Scholar [29] A. V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posednessin the space of continuous functions,, Nonlinear Analysis: Theory, 70 (2009), 3978.  doi: 10.1016/j.na.2008.08.006.  Google Scholar [30] A. V. Rezounenko, Non-linear partial differential equations with discrete state-dependent delays in a metric space,, Nonlinear Analysis: Theory, 73 (2010), 1707.  doi: 10.1016/j.na.2010.05.005.  Google Scholar [31] R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,", AMS, 49 (1997).   Google Scholar [32] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Annali di Mat. Pura ed Appl., 146 (1987), 65.   Google Scholar [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.  doi: 10.1098/rspa.2001.0789.  Google Scholar [34] J. W. H. So and Y. Yang, Dirichlet problem for the diffusive Nicholson's blowflies equation,, J. Differential Equations, 150 (1998), 317.   Google Scholar [35] R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar [36] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations,, Transactions of AMS, 200 (1974), 395.   Google Scholar [37] H. O. Walther, Stable periodic motion of a system with state-dependent delay,, Differential and Integral Equations, 15 (2002), 923.   Google Scholar [38] H. O. Walther, The solution manifold and C1-smoothness for differential equations with state-dependent delay,, J. Differential Equations, 195 (2003), 46.  doi: 10.1016/j.jde.2003.07.001.  Google Scholar [39] H. O. Walther, On a model for soft landing with state-dependent delay,, J. Dynamics and Differential Eqs, 19 (2007), 593.  doi: 10.1007/s10884-006-9064-8.  Google Scholar [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.  doi: 10.1007/s10884-010-9168-z.  Google Scholar [41] X. Wang and Z. Li, Dynamics for a type of general reaction-diffusion model,, Nonlinear Analysis, 67 (2007), 2699.  doi: 10.1016/j.na.2006.09.034.  Google Scholar [42] E. Winston, Uniqueness of the zero solution for differential equations with state-dependence,, J. Differential Equations, 7 (1970), 395.  doi: 10.1016/0022-0396(70)90118-X.  Google Scholar [43] J. Wu, "Theory and Applications of Partial Functional Differential Equations,", Springer-Verlag, (1996).   Google Scholar [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.  doi: 10.1007/s00033-010-0112-1.  Google Scholar [45] K. Yosida, "Functional Analysis,", Springer-Verlag, (1965).   Google Scholar
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