The stack exchange solutions of
https://mathoverflow.net/questions/172172/finite-speed-of-propagation-of-wave-equation
Use energy method prove wave equation has a unique of solution.
use different definitions of $E'(t)$, where in one case the boundary term is missing.
I am following
(Graduate Studies in Mathematics, V. 19 GSM_19) Lawrence C. Evans-Partial Differential Equations-American Mathematical Society (1998)
and in that book, the reasoning to discard the boundary term given the conditions that $w_t=w=0$ on the boundary are:
But there is no explanation of why $\nabla w$ should be zero on the boundary in order to make the integral of the energy over the boundary zero.