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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 |
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. |
show all references
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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