Strongly continuous semigroups, weak solutions, and the variation of constants formula
J. M. Ball
Proc. Amer. Math. Soc. 63 (1977), 370-373
Primary 47D05; Secondary 34G05
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Abstract: Let A be a densely defined closed linear operator on a Banach space X, and let . A definition of weak solutions of the equation 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.
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