about the proof of existence of weak solution of linear hyperbolic equation ( via Galerkin method)

Question

In the classical PDE book of evans in the chapter 7 he proves using Galerkin method the existence of weak solution to the linear hyperbolic equation. In the page 385 (edition of 1998 ) he writes

Integrating by parts twice with respect to $t$ the equation

$$ \int_{0}^{T} \langle u'', v\rangle + B[u,v;t] \ dt = \int_{0}^{T} (f,v) \ dt$$ we obtain :

$$ \int_{0}^{T} \langle v'', u\rangle + B[u,v;t] \ dt \int_{0}^{T} (f,v) \ dt - (u(0), v'(0)) + \langle u' (0), v(0)\rangle$$.

for $v \in C^{2}([0,T]; H^{1}_{0}(U))$ with $v(T) = v'(T) = 0$

How to prove this "integration by part formula"? Someone can give me a hint ? Thanks in advance

Answer 1

Try this one Theorem 6.40 or this pages 181-182. You could try also type Bochner Integral on google or Bochner Intgration Parts. I did it and I found a lot of things.