I am reading the book of Evans, Partial differential Equations ... wave equation section 2.4; subsection 2.4.3: Energy methods. Arriving at the theorem:
Theorem 5 (Uniqueness for wave equation). There exists at most one function $u \in C^{2}(\overline{U}_{T})$ solving
$u_{tt} -\Delta u=f $ in $ U_{T}$
$u=g $ on $ \Gamma_{T}$
$u_{t}=h$ on $U \times \{t=0\}.$
Proof. If $\tilde{u}$ is another such solution, then $ w:=u-\tilde{u}$ solves
$w_{tt} -\Delta w=0 $ in $ U_{T}$
$w=0 $ on $ \Gamma_{T}$
$w_{t}=0$ on $U \times \{t=0\}.$
Define the "energy"
$e(t):=\frac{1}{2} \int_{U} w^{2}_{t}(x,t)+ \mid Dw(x,t)\mid ^{2} dx (0\leq t \leq T).$
We compute
$\dot{e}(t)=\int_{U} w_{t}w_{tt}+ Dw \cdot Dw_{t}dx (\cdot = \frac{d}{dt})$
$=\int_{U}w_{t}(w_{tt} - \Delta w)dx=0$.
There is no boundary term since $w=0$, and hence $w_{t}=0$, on $\partial U \times [0,T].$ Thus for all $0\leq t \leq T, e(t)= e(0)=0$, and so $w_{t}, Dw \equiv 0$ within $U_{T}$. Since $w \equiv 0$ on$ U \times \{t=0\}$, we conclude $w=u-\tilde {u}\equiv 0$ in $U_{T}$.
I have two questions:
1) What is the motivation for the definition of $e(t)$
2)$\int_{U} w_{t}w_{tt}+ Dw \cdot Dw_{t}dx $
$=\int_{U}w_{t}(w_{tt} - \Delta w)dx$. How to justify this equality?
Thank very much.