First, you should have $\varphi^2$ inside all of the integrals instead of just $\varphi$. For the equivalence, you can read Theorem 2.44 of Coron, Control and Nonlinearity or Theorem 11.2.1 of Tucsnak and Weiss, Observation and Control for Operator Semigroups. Basically, this is Integration by parts, Riesz Theorem in Hilbert spaces and Closed Graph Theorem. I can put a full answer if tou want, but I can't promise it will we easier to understand than these references :) .