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

Abstract

Let A be a densely defined closed linear operator on a Banach space X, and let f ∈ L 1 ( 0 , τ ; X ) f \in {L^1}(0,\tau ;X) . A definition of weak solutions of the equation u ˙ = A u + f ( t ) \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.