Wave Equation, Energy methods.

Question

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.

Answer 1

From comments of Yimin, more-or-less:

2) Integrate by parts. See Appendix C, theorem 3, part (ii) in Evans. Note that the boundary term vanishes due to the assumptions.

1) Energy, in PDE, often means (the integral of) a quantity you can minimise in order to solve the equation in question, by which I mean that the function that minimises the energy also solves the PDE. This often corresponds to energy as it is used in physics.