Energy Estimate for Nonlinear Wave Equation in $H^s$ Space In Theorem 2.1 of Prof. Lawrie notes (which can be found here), he provided without proof the following result:

Let $(f,g)\in H^s\times H^{s-1}$ and $h\in L^1([0,T], H^{s-1})$. Then the linear wave equation
$$\begin{cases}
\Box u = h \\
u(0,x) = f(x)\\
\partial_tu(0,x) = g(x)
\end{cases}$$
has a unique solution $u\in C([0,T],H^s)\cap C^1([0,T],H^{s-1})$ which satisfies, for each $0\le t\le T$, the following energy estimates:
$$
\|u(t,\cdot)\|_{H^s} + \|\partial_t u(t,\cdot)\|_{H^{s-1}} \le C(1+t)\left(\|f\|_{H^s} + \|g\|_{H^{s-1}} + \int_0^t\|h(\tau,\cdot)\|_{H^{s-1}} d\tau\right)
$$

To this theorem, I have three questions:

*

*I am not too worried about the uniqueness but I have no idea how to proof the inequality. In particular, how should I bound $\|u(t,\cdot)\|_{H^s}$? I've tried the energy method with $E(t):=\frac{1}{2}\int_{\mathbb{R}^n}|u(t,x)|^2\,dx$ but I didn't have much success with this. How do I make the $(1+t)$ appear when bounding this term?


*When bounding $\|\partial_t u(t,\cdot)\|_{H^{s-1}}$, I used integration by parts and simply discarded the boundary term; am I allowed to do that? The working for that portion looks like this
\begin{align}
\int_{\mathbb{R}^n} (\partial u)\cdot(\partial_t\partial u)dx
&= \int_{\mathbb{R}^n} \left[(\partial_t u)(\partial_t^2 u) + \sum_{i=1}^n (\partial_i u)(\partial_i \partial_t u)\right]dx \\
&= \int_{\mathbb{R}^n} \left[(\partial_t u)(\partial_t^2 u) - \sum_{i=1}^n (\partial_i^2 u)(\partial_t u)\right]dx 
\end{align}


*What is going on with all the spaces here? What is $h\in L^1([0,T], H^{s-1})$? Isn't the function $h$ suppose to be $\mathbb{R}^{1+n}\to\mathbb{R}$? Then how can it be a map of $[0,T]\to H^{s-1}(\mathbb{R^n})$? And what does $u\in C([0,T],H^s)\cap C^1([0,T],H^{s-1})$ mean? How do you even intersect these two spaces? I'm really perplexed by this so please help shed some light here. (I know this post, to some extent, discuss this but whatever is going on there makes no sense to me as well)
Any help would be appreciated; thanks in advance.
 A: For the first part of the third question, although the function $h=h(t,x)$ is from $ [0,T]\times \mathbb{R}^n$ to $\mathbb{R}$. One can fix a $t\in [0,T]$ to regard $h(t,\cdot)$ as a function of $x\in\mathbb{R}$. Varying $t\in [0,T]$, we get a map from $[0,T]$ to $H^{s-1}(\mathbb{R}^n)$. So basically we can regard the function $h$ as a map:$[0,T]\to H^{s-1}(\mathbb{R}^n)$.
In this way, one can define the space $L^1([0,T];H^{s-1}(\mathbb{R}^n))$ by
$$
 \| h \|_{L^1([0,T];H^{s-1}(\mathbb{R}^n))}:= \int^T_0 \| h(t,\cdot) \|_{H^{s-1}(\mathbb{R}^n)}\,dt <\infty.
$$
For the second part, similarly, $u \in C([0,T],H^s)$ means that we regard $u$ as a map $t\to u(t,\cdot)$ and this map is continuous and lies in the space $C([0,T],H^s)$ with
$$
 \| u \|_{C([0,T];H^{s}(\mathbb{R}^n))}:= \max_{0\leq t \leq T} \| u(t,\cdot) \|_{H^{s}(\mathbb{R}^n)} <\infty.
$$
To define the space $C^1([0,T],H^{s-1})$, it requires the map $t\to u(t,\cdot)$ is continuously differentiable. The norm of $C^1([0,T],H^{s-1}(\mathbb{R}^n))$ is therefore given by
$$
 \| u\|_{C^1([0,T],H^{s-1}(\mathbb{R}^n))}:= \| u \|_{C([0,T],H^{s-1}(\mathbb{R}^n))}+\| u'\|_{C([0,T],H^{s-1}(\mathbb{R}^n))},
$$
where $u'$ denotes the Fréchet derivative of the map $t \to u(t,\cdot)$. Of course, $u\in  C([0,T],H^s)\cap C^1([0,T],H^{s-1})$ means $u\in C([0,T],H^s)$ and $u\in C^1([0,T],H^{s-1})$.
