Non - autonomous inhomogeneous boundary cauchy problems and retarded equations
DOI:
https://doi.org/10.4067/S0716-09172003000200005Keywords:
Boundary Cauchy problem, Evolution families, Classical solution, Well-posedness, Variation of constants formula, Retarded equation.Abstract
In this paper we prove the existence and the uniqueness of the classical solution of non-autonomous inhomogeneous boundary Cauchy problems, and that this solution is given by a variation of constants formula. This result is applied to show the existence of solutions of a retarded equation.
References
[1] K. J. Engel and R. Nagel: One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194, SpringerVerlag 2000.
[2] G. Greiner: Perturbing the boundary conditions of a generator, Houston J. Math. 13, pp. 213-229, (1987).
[3] G. Greiner: Semilinear boundary conditions for evolution equations of hyperbolic type, Lecture Notes in Pure and Appl. Math. 116 Dekker, New York, pp. 127-153, (1989).
[4] R. Grimmer and E. Sinestrari: Maximum norm in one-dimensional hyperbolic problems, Diff. Integ. Equat. 5, pp. 421-432, (1992).
[5] T. Kato: Linear evolution equations of ”hyperbolic” type, J. Fac. Sci. Tokyo, 17, pp. 241-258, (1970).
[6] H. Kellermann: Linear evolution equations with time dependent domain, Semesterbericht Funktionalanalysis T¨ubingen, Wintersemester, pp. 15-44, (1985/1986).
[7] N. T. Lan: On nonautonomous Functional Differential Equations, J. Math. Anal. Appl. 239, pp. 158-174, (1999).
[8] N. T. Lan and G. Nickel: Time-Dependent operator matrices and inhomogeneous Cauchy Problems, Rend. Circ. Mat. Palermo XLVII, pp. 5-24, (1998).
[9] R. Nagel and E. Sinestrari: Inhomogeneous Volterra integrodifferential equations for Hille-Yosida operators, Marcel Dekker, Lecture Notes Pure Appl. Math. 150, pp. 51-70, (1994).
[10] A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations, New York Springer (1983).
[11] N. Tanaka: Quasilinear evolution equations with non-densely defined operators, Diff. Int. Eq. 9, pp. 1067-1106, (1996).
[2] G. Greiner: Perturbing the boundary conditions of a generator, Houston J. Math. 13, pp. 213-229, (1987).
[3] G. Greiner: Semilinear boundary conditions for evolution equations of hyperbolic type, Lecture Notes in Pure and Appl. Math. 116 Dekker, New York, pp. 127-153, (1989).
[4] R. Grimmer and E. Sinestrari: Maximum norm in one-dimensional hyperbolic problems, Diff. Integ. Equat. 5, pp. 421-432, (1992).
[5] T. Kato: Linear evolution equations of ”hyperbolic” type, J. Fac. Sci. Tokyo, 17, pp. 241-258, (1970).
[6] H. Kellermann: Linear evolution equations with time dependent domain, Semesterbericht Funktionalanalysis T¨ubingen, Wintersemester, pp. 15-44, (1985/1986).
[7] N. T. Lan: On nonautonomous Functional Differential Equations, J. Math. Anal. Appl. 239, pp. 158-174, (1999).
[8] N. T. Lan and G. Nickel: Time-Dependent operator matrices and inhomogeneous Cauchy Problems, Rend. Circ. Mat. Palermo XLVII, pp. 5-24, (1998).
[9] R. Nagel and E. Sinestrari: Inhomogeneous Volterra integrodifferential equations for Hille-Yosida operators, Marcel Dekker, Lecture Notes Pure Appl. Math. 150, pp. 51-70, (1994).
[10] A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations, New York Springer (1983).
[11] N. Tanaka: Quasilinear evolution equations with non-densely defined operators, Diff. Int. Eq. 9, pp. 1067-1106, (1996).
Published
2017-04-24
How to Cite
[1]
M. Filali and M. Moussi, “Non - autonomous inhomogeneous boundary cauchy problems and retarded equations”, Proyecciones (Antofagasta, On line), vol. 22, no. 2, pp. 145-159, Apr. 2017.
Issue
Section
Artículos
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.