In J. L. - Lions book Quelques méthodes de résolution des problèmes aux limites non linéaires, the author proves the following lemma:

Lemme 6.1: Let w be a function satisfying $w \in L^\infty(0,T;V)$, $w' \in L^\infty(0,T;H)\cap L^p(0,T;L^p(\Omega))$, \begin{equation} \left\{\begin{array}{lc} w''+Aw=g, & g \in L^2(0,T;L^2(\Omega))+L^{p'}(0,T;L^{p'}(\Omega)),\\ w(0)=u_0, w'(0)=u_1 \end{array}\right. \end{equation} Then, for almost all $t \in [0, T]$, we have the following energy inequality: $$a(w(t),w(t))+|w'(t)|^2\geq a(u_0,u_0)+|u_1|^2+2\int_0^t (g,w')d\sigma, $$ where $p^{-1}+{p'}^{-1}=1.$

Under the additional hypothesis that $g \in L^2(0,T;H)$, it is proved in the book Lions, J. L., Magenes, E. Non-Homogeneous Boundary Value Problems And Applications I. (Lemma 8.3) that equality holds.

My question: I would like to know if it is possible to prove that in the Lemme 6.1 hypotheses equality holds. If not, is there any counterexample for this?