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A class of linear partial neutral functional differential equations with nondense domain. (English) Zbl 0915.35109

The paper deals with the linear neutral differential equations on a Banach space \(X\), \[ {d\over dt} (x(t)- Bx(t- h))= A_0x(t)+ Cx(t- h)+ L(x_t),\quad t\geq 0,\quad x(0)= \varphi\in C_X, \] where \(A_0: D(A_0)\subseteq X\mapsto X\) is a linear operator, \(B,C\in{\mathcal L}(X)\), \(L\) is a continuous linear functional from \(C_X:= C([-h,0],X)\) into \(X, \varphi\in C_X\) and \(x_t(\theta)= x(t+ \theta)\), \({-h\leq\theta\leq 0}\). A variation-of-constants formula is derived allowing transformation of integral solutions of the above problem to solutions of an abstract Volterra integral equation. The authors prove existence and uniqueness results and show that the solutions generate an integrated semigroup.
Reviewer: D.Bainov (Sofia)

MSC:

35R10 Partial functional-differential equations
35K40 Second-order parabolic systems
35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
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[1] Adimy, M., Bifurcation de Hopf locale par les semi-groupes intégrés, C.R. Acad. Sci. Paris, 311, 423-428 (1990) · Zbl 0705.34079
[2] Adimy, M., Integrated semigroups and delay differential equations, J. Math. Anal. Appl., 177, 125-134 (1993) · Zbl 0776.34045
[4] Adimy, M.; Arino, O., Bifurcation de Hopf globale pour des équations à retard par des semi-groupes intégrés, C.R. Acad. Sci. Paris, 317, 767-772 (1993) · Zbl 0783.34054
[5] Adimy, M.; Ezzinbi, K., Equations de type neutre et semi-groupes intégrés, C.R. Acad. Sci. Paris, 318, 529-534 (1994) · Zbl 0802.34078
[6] Adimy, M.; Ezzinbi, K., Semi-groupes intégrés et équations différentielles à retard en dimension infinie, C.R. Acad. Sci. Paris, 323, 481-486 (1996) · Zbl 0859.34062
[7] Arendt, W., Resolvent positive operators and integrated semigroup, Proc. London Math. Soc. 3, 54, 321-349 (1987) · Zbl 0617.47029
[8] Arendt, W., Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59, 327-352 (1987) · Zbl 0637.44001
[9] Arino, O.; Sanchez, E., Linear theory of abstract functional differential equations of retarded type, J. Math. Anal. Appl., 191, 547-571 (1995) · Zbl 0821.34074
[10] Bellman, R.; Cooke, K., Differential Difference Equations (1963), Academic Press: Academic Press San Diego
[11] Busenberg, S.; Wu, B., Convergence theorems for integrated semigroups, Differential Integral Equations, 5, 509-520 (1992) · Zbl 0786.47036
[12] Cooke, K. L.; Krumme, D. W., Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations, J. Math. Anal. Appl., 24, 372-387 (1968) · Zbl 0186.16902
[13] Cruz, M. A.; Hale, J. K., Asymptotic behavior of neutral functional differential equations, Arch. Rational Mech. Anal., 34, 331-353 (1969) · Zbl 0211.12301
[14] Da Prato, G.; Sinestrari, E., Differential operators with non-dense domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14, 285-344 (1987) · Zbl 0652.34069
[15] Datko, R., Linear autonomous neutral differential equations in a Banach space, J. Differential Equations, 25, 258-274 (1977) · Zbl 0402.34066
[17] Fitzgibbon, W. E., Semilinear functional differential equations in Banach space, J. Differential Equations, 29, 1-14 (1978) · Zbl 0392.34041
[18] Grabosch, A.; Moustakas, U., A semigroup approach to retarded differential equations, One-Parameter Semigroups of Positive Operators. One-Parameter Semigroups of Positive Operators, Lecture Notes in Math., 1184 (1986), Springer-Verlag: Springer-Verlag Berlin/New York, p. 219-232
[19] Hale, J. K., A class of neutral equations with the fixed-point property, Proc. Nat. Acad. Sci. U.S.A., 67, 136-137 (1970) · Zbl 0207.45403
[20] Hale, J. K., Critical cases for neutral functional differential equations, J. Differential Equations, 10, 59-82 (1971) · Zbl 0223.34057
[21] Hale, J. K., Theory of Functional Differential Equations (1977), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0425.34048
[22] Hale, J. K., Partial neutral functional differential equations, Rev. Roumaine Math. Pures Appl., 39, 339-344 (1994) · Zbl 0817.35119
[23] Hale, J. K., Coupled oscillators on a circle, Resenhas, 1, 441-457 (1994) · Zbl 0857.35127
[24] Hale, J. K.; Lin, X. B.; Raugel, G., Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp., 50, 89-123 (1988) · Zbl 0666.35013
[25] Hale, J. K.; Lunel, S., Introduction to Functional Differential Equations (1993), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0787.34002
[26] Hale, J. K.; Meyer, K. R., A class of functional equations of neutral type, Mem. Amer. Math. Soc., 76 (1967) · Zbl 0179.20501
[27] Henry, D., Linear autonomous neutral functional differential equations, J. Differential Equations, 15, 106-128 (1974) · Zbl 0294.34047
[28] Hieber, M., Integrated Semigroups and Differential Operators on\(L^p\) (1989) · Zbl 0697.47038
[29] Kato, T., Perturbation Theory for Linear Operators (1966), Springer-Verlag: Springer-Verlag Berlin · Zbl 0148.12601
[30] Kunisch, K.; Schappacher, W., Variation of constants formula for partial differential equations with delay, Nonlinear Anal., 5, 123-142 (1981) · Zbl 0458.35091
[31] Kunisch, K.; Schappacher, W., Necessary conditions for partial differential equations with delay to generate a \(C_0\), J. Differential Equations, 50, 49-79 (1983) · Zbl 0533.35082
[32] Kellermann, H., Integrated Semigroups (1986) · Zbl 0604.47025
[33] Kellermann, H.; Hieber, M., Integrated semigroup, J. Funct. Anal., 15, 160-180 (1989)
[34] Memory, M., Invariant manifolds for partial functional differential equations, (Arino, O.; Axelrod, D. E.; Kimmel, M., Mathematical Population Dynamics (1991), Dekker), 223-232 · Zbl 0808.35169
[35] Neubrander, F., Integrated semigroups and their applications to the abstract Cauchy problems, Pacific J. Math., 135, 111-155 (1988) · Zbl 0675.47030
[36] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0516.47023
[37] Sinestrari, E., On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl., 107, 16-66 (1985) · Zbl 0589.47042
[38] Thieme, H., Integrated semigroups and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152, 416-447 (1990) · Zbl 0738.47037
[39] Travis, C. C.; Webb, G. F., Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200, 395-418 (1974) · Zbl 0299.35085
[40] Webb, G. F., Asymptotic stability for abstract functional differential equations, Proc. Amer. Math. Soc., 54, 225-230 (1976) · Zbl 0324.34079
[41] Wu, J., Theory and Applications of Partial Functional Differential Equations (1996), Springer-Verlag: Springer-Verlag Berlin/New York
[42] Wu, J.; Xia, H., Self-sustained oscillations in a ring array of coupled lossless transmission lines, J. Differential Equations, 124, 247-278 (1996) · Zbl 0840.34080
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