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Denote by $\\A(t) \\in \\L(V,V')$ the associated operator. Given $f \\in L^2(0,T, V')$, one knows that for each $u_0 \\in H$ there is a unique solution $u\\in H^1(0,T;V')\\cap L^2(0,T;V)$ of $$\\dot u(t) + \\A(t) u(t) = f(t), \\, \\, u(0) = u_0.$$ %\\begin{align*} %&\\dot u(t) + \\A(t)u(t)= f(t)\\ %& u(0)=u_0. %\\end{align*} This result by J. L. Lions is well-known. 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