We prove two general theorems related to the smooth dependence on data of mild solutions to evolution Cauchy problems and provide some of their applications to the Faedo-Galerkin method for approximating solutions as well as to the existence and uniqueness of periodic solutions.
@article{BUMI_2009_9_2_3_731_0,
author = {Giovanni Vidossich},
title = {Smooth Dependence on Initial Data of Mild Solutions to Evolution Equations},
journal = {Bollettino dell'Unione Matematica Italiana},
volume = {2},
year = {2009},
pages = {731-754},
zbl = {1186.65066},
mrnumber = {2569301},
language = {en},
url = {http://dml.mathdoc.fr/item/BUMI_2009_9_2_3_731_0}
}
Vidossich, Giovanni. Smooth Dependence on Initial Data of Mild Solutions to Evolution Equations. Bollettino dell'Unione Matematica Italiana, Tome 2 (2009) pp. 731-754. http://gdmltest.u-ga.fr/item/BUMI_2009_9_2_3_731_0/
[1] , Periodic solutions of semilinear equations of evolution of compact type, J. Math. Anal. Appl., 82 (1981), 33-48. | MR 626739 | Zbl 0465.34014
[2] - , Results on periodic solutions of parabolic equations suggested by elliptic theory, Boll. U.M.I., 1-B (1982), 1089-1104. | MR 683495 | Zbl 0501.35005
[3] - , An Introduction to Semilinear Evolution Equations, Clarendon Press, Oxford, 1998. | MR 1691574 | Zbl 0926.35049
[4] - , Stable manifolds for hyperbolic sets, In: "Proc. Symp. Pure Math.", vol. XIV, Amer. Math. Society (Providence, 1970), 133-163. | MR 271991 | Zbl 0215.53001
[5] , A class of semi-linear equations of evolution, Israel J. Math., 20 (1975), 23-36. | MR 374996 | Zbl 0305.47022
[6] , Smooth dependence of solutions of differential equations on initial data: a simple proof, Bol. Soc. Brasil. Mat., 4 (1973), 55-59. | MR 361238 | Zbl 0337.34005
[7] , Infinite-dimensional Dynamical Systems in Mechanics and Physics (Springer-Verlag, New York, 1988). | MR 953967 | Zbl 0662.35001
[8] , Continuous dependence for parabolic evolution equations, In: "Recent Trends in Differential Equations", (ed.), World Scientific (Singapore, 1992), 559-568. | MR 1180138 | Zbl 0832.34058
[9] , A (semi-encyclopedic) first course in ODEs, in a "never-ending" preparation since 1974.
[10] , Boundary value problems for differential equations in Banach space, J. Math. Anal. Appl., 70 (1979), 589-598. | MR 543596 | Zbl 0438.34057