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Nonuniqueness and wellposedness of abstract Cauchy problems in a Fréchet space

Published online by Cambridge University Press:  17 April 2009

Peer Christian Kunstmann
Peer Christian Kunstmann
Affiliation:
Mathematisches Institut I, Universität Karlsruhe, Englerstr. 2, 76128 Karlsrube, Germany, e-mail: [email protected]
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Abstract

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Suppose that A is a closed linear operator in a Fréchet space X. We show that there always is a maximal subspace Z containing all x ∈ X for which the abstract Cauchy problem has a mild solution, which is a Fréchet space for a stronger topology. The space Z is isomorphic to a quotient of a Fréchet space F, and the part Az of A in Z is similar to the quotient of a closed linear operator B on F for which the abstract Cauchy problem is well-posed. If mild solutions of the Cauchy problem for A in X are unique it is not necessary to pass to a quotient, and we reobtain a result due to R. deLaubenfels.

Moreover, we obtain a continuous selection operator for mild solutions of the inhomogeneous equation.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 63 , Issue 1 , February 2001 , pp. 123 - 131
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Bourbaki, N., Topological vector spaces (Springer-Verlag, Berlin, Heidelberg, New York, 1987).CrossRefGoogle Scholar
[2]deLaubenfels, R., ‘C-semigroups and strongly continuous semigroups’, Israel J. Math. 81 (1993), 227–255.CrossRefGoogle Scholar
[3]deLaubenfels, R., ‘Automatic well-posedness with the abstract Cauchy problem on a Frechet space’, J. London Math. Soc. Ser. 2 48 (1993), 526–536.CrossRefGoogle Scholar
[4]deLaubenfels, R., Existence families, functional calculi and evolution equations, Lecture Notes Maths 1570 (Springer-Verlag, Berlin, Heidelberg, New York, 1994).CrossRefGoogle Scholar
[5]Herzog, G. and Lemmert, R., ‘Nonlinear fundamental systems for linear differential equations in Fréchet spaces’, Demonstratio Math. 33 (2000), 313–318.Google Scholar
[6]Kantorovitz, S., ‘The Hille-Yosida space of an arbitrary operator’, J. Math. Anal. Appl. 136 (1988), 107–111.CrossRefGoogle Scholar
[7]Köthe, G., Topological vector spaces II (Springer-Verlag, New York, Heidelberg, Berlin, 1979).CrossRefGoogle Scholar
[8]Lyubich, Yu.I., ‘Conditions for the uniqueness of the solution to Cauchy's abstract problem’, Soviet Math. Dokl. 1 (1960), 110–113.Google Scholar
[9]Rudin, W., Real and complex analysis, Series in Higher Mathematics (McGraw-Hill, New York, Dusseldorf, Johannesburg, 1987).Google Scholar
[10]Meise, R. and Taylor, B.A., ‘Whitney's extension theorem in spaces of ultradifferentiable functions of Beurling type’, Ark. Mat. 26 (1988), 265–287.CrossRefGoogle Scholar