Strongly continuous semigroups, weak solutions, and the variation of constants formula
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- by J. M. Ball PDF
- Proc. Amer. Math. Soc. 63 (1977), 370-373 Request permission
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
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J. M. Ball, On the asymptotic behaviour of generalized processes, with applications to nonlinear evolution equations, J. Differential Equations (to appear).
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- A. V. Balakrishnan, Applied functional analysis, Applications of Mathematics, No. 3, Springer-Verlag, New York-Heidelberg, 1976. MR 0470699
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 63 (1977), 370-373
- MSC: Primary 47D05; Secondary 34G05
- DOI: https://doi.org/10.1090/S0002-9939-1977-0442748-6
- MathSciNet review: 0442748