×

A survey on new methods for partial functional differential equations and applications. (English) Zbl 1463.35486

Summary: This work is a survey of many papers dealing with new methods to study partial functional differential equations. We propose a new reduction method of the complexity of partial functional differential equations and its applications. Since, any partial functional differential equation is well-posed in a infinite dimensional space, this presents many difficulties to study the qualitative analysis of the solutions. Here, we propose to reduce the dimension from infinite to finite. We suppose that the undelayed part is not necessarily densely defined and satisfies the Hille-Yosida condition. The delayed part is continuous. We prove the dynamic of solutions are obtained through an ordinary differential equations that is well-posed in a finite dimensional space. The powerty of this results is used to show the existence of almost automorphic solutions for partial functional differential equations. For illustration, we provide an application to the Lotka-Volterra model with diffusion and delay.

MSC:

35R10 Partial functional-differential equations
47A10 Spectrum, resolvent
47D06 One-parameter semigroups and linear evolution equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adimy, M.; Ezzinbi, K.; Laklach, M., Spectral decomposition for partial neutral functional differential equations, Can. Appl. Math. Q., 9, 1, 1-34 (2001) · Zbl 1112.34341
[2] Travis, Cc; Webb, Gf, Existence and stability for partial functional differential equations, Trans. Am. Math. Soc., 200, 395-418 (1974) · Zbl 0299.35085 · doi:10.1090/S0002-9947-1974-0382808-3
[3] Wu, J., Theory and Applications of Partial Functional Differential Equations (1996), Berlin: Springer, Berlin · Zbl 0870.35116
[4] Bochner, S., Continuous mappings of almost automorphic and almost automorphic functions, Proc. Natl. Sci. USA, 52, 907-910 (1964) · Zbl 0134.30102 · doi:10.1073/pnas.52.4.907
[5] N’Guérékata, Gm, Almost Automorphic and Almost Automorphic Functions in Abstract Spaces (2001), Amesterdam: Kluwer, Amesterdam · Zbl 1001.43001
[6] N’Guérékata, Gm, Almost auotmorphy, almost periodicity and stability of motions in Banach spaces, Forum Maths, 13, 581-588 (2001) · Zbl 0974.34058
[7] Hino, Y.; Murakami, S., Almost automorphic for abstract functional differential equations, J. Math. Anal. Appl., 286, 741-752 (2003) · Zbl 1046.34088 · doi:10.1016/S0022-247X(03)00531-6
[8] Adimy, M.; Ezzinbi, K., Existence and linearized stability for partial neutral functional differential equations, Differ. Equ. Dyn. Syst., 7, 371-417 (1999) · Zbl 0983.34075
[9] Arendt, W.; Batty, Cjk; Hieber, M.; Neubrander, F., Vector Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics (2001), Basel: Birkhäuser, Basel · Zbl 0978.34001
[10] Hale, J.; Kato, J., Phase spaces for retarded equations with unbounded delay, Funkc. Ekvac, 21, 11-41 (1978) · Zbl 0383.34055
[11] Adimy, M.; Bouzahir, H.; Ezzinbi, K., Local Existence and stability for some partial functional differential equations with infinite delay, Nonlinear Anal. Theory Methods Appl., 48, 323-348 (2002) · Zbl 0996.35080 · doi:10.1016/S0362-546X(00)00184-X
[12] Hino, Y.; Murakami, S.; Naito, T.; Minh, Nv, A variation of constants formula for abstract functional differential equations in the phase spaces, J. Differ. Equ., 179, 336-355 (2002) · Zbl 1005.34070 · doi:10.1006/jdeq.2001.4020
[13] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences (1983), New York: Springer, New York · Zbl 0516.47023
[14] Thieme, Hr, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differ. Integral Equ., 3, 6, 1035-1066 (1990) · Zbl 0734.34059
[15] Engel, Kj; Nagel, R., One-Parameter Semigroups of Positive Operators. Lecture Notes in Mathematics (1986), Berlin: Springer, Berlin · Zbl 0585.47030
[16] Zeidler, E., Nonlinear Functional Analysis and It’s Applications, Tome I, Fixed Point Theorem (1993), Berlin: Springer, Berlin · Zbl 0794.47033
[17] Hino, Y.; Murakami, S.; Naito, T., Functional Differential Equations with Infinite Delay. Lectures Notes in Mathematics (1991), Berlin: Springer, Berlin · Zbl 0732.34051
[18] Benkhalti, R.; Bouzahir, H.; Ezzinbi, K., Existence of Periodic solutions for some partial functional differential equations with infinite delay, J. Math. Anal. Appl., 256, 257-280 (2001) · Zbl 0981.35093 · doi:10.1006/jmaa.2000.7321
[19] Fink, A., Almost Periodic Differential Equations, Lectures Notes (1974), New York: Springer, New York · Zbl 0325.34039
[20] Da Prato, G.; Sinestrari, E., Differential operators with nondense domains, Ann. Sc. Norm. Super. Pisa, 14, 2, 285-344 (1987) · Zbl 0652.34069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.