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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: peer.kunstmann@math.uni-karlsruhe.de
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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

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