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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Strongly continuous semigroups, weak solutions, and the variation of constants formula


Author: J. M. Ball
Journal: Proc. Amer. Math. Soc. 63 (1977), 370-373
MSC: Primary 47D05; Secondary 34G05
MathSciNet review: 0442748
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Abstract: Let A be a densely defined closed linear operator on a Banach space X, and let $ f \in {L^1}(0,\tau ;X)$. A definition of weak solutions of the equation $ \dot u = Au + f(t)$ is given. It is shown that a necessary and sufficient condition for the existence of unique weak solutions for every initial data in X is that A generate a strongly continuous semigroup on X, and that in this case the solution is given by the variation of constants formula.


References [Enhancements On Off] (What's this?)

  • [1] J. M. Ball, On the asymptotic behaviour of generalized processes, with applications to nonlinear evolution equations, J. Differential Equations (to appear).
  • [2] Seymour Goldberg, Unbounded linear operators: Theory and applications, McGraw-Hill Book Co., New York, 1966. MR 0200692 (34 #580)
  • [3] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473 (34 #3324)
  • [4] A. V. Balakrishnan, Applied functional analysis, Springer-Verlag, New York, 1976. Applications of Mathematics, No. 3. MR 0470699 (57 #10445)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0442748-6
PII: S 0002-9939(1977)0442748-6
Keywords: Linear semigroups, weak solutions, variation of constants formula
Article copyright: © Copyright 1977 American Mathematical Society