I've been struggling for a while to understand the proof in Evans of the Energy Estimates of Linear Hyperbolic PDE. This is Theorem 2 in section 7.2. I have added the proof from Evans for convenience. I'm not a mathematician, rather an economist looking to improve my maths skills. Sorry if this is a rather trivial question for this website. Any advice would be very much appreciated. My problem is understanding the following:
Consider \begin{equation} B_1=\int_U \sum_{i,j=1}^n a^{ij}(\mathbf{u}_m)_{x_i}(\mathbf{u}'_m)_{x_j}\;dx \tag{1} \end{equation} We aim to write this expression as \begin{equation} B_1=\frac{d}{dt}\bigg(\frac{1}{2}A[\mathbf{u}_m,\mathbf{u}_m;t]\bigg)-\frac{1}{2}\int_U\sum_{i,j=1}^n a_t^{ij}(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}\;dx \tag{2}\end{equation} since $a^{ij}=a^{ji}$ and $A[u,v;t]$ is given by \begin{equation} A[u,v;t]:=\int_U \sum_{i,j=1}^n a^{ij}u_{x_i}v_{x_j}\;dx\;\;\;(u,v\in H^1_0(U)) \tag{3}\end{equation} Then from our new expression for $B_1$ we find that \begin{equation} B_1\geq \frac{d}{dt}\bigg(\frac{1}{2}A[\mathbf{u}_m,\mathbf{u}_m;t]\bigg)-C\|\mathbf{u}_m\|^2_{H^1_0(U)} \tag{4}\end{equation} My problem is that I can't see how to get from $(1)$ to $(2)$. It seems like it's just a case of bringing the differentiation operator through the integral and summation as usual but I just can't get it to work. My second question is how to get the inequality $(4)$. In particular I'm not sure how to bound the second term on the right hand side of $(2)$.
I have tried the following idea \begin{align} \int_U\sum_{i,j=1}^n a_t^{ij}(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}\;dx &\leq C\int_U\sum_{i,j=1}^n(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}\;dx \\&= C\int_U\sum_{i=1}^n|(\mathbf{u}_m)_{x_i}|^2+\sum_{i\neq j}^n(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}\;dx \tag{5} \end{align} But I get stuck here as I don't know the sign of $(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}$ over $U$
$\mathbf{Edit}$ 6/8/17 I have realised that I have forgotten to mention that $\mathbf{u}_m(t)=\sum d^k_m(t)w_k$ where $\{w\}_{k=1}^\infty$ is an orthogonal basis of $H^1_0(U)$ and $L^2(U)$ from our Galerkin approximations. From this can I conclude that the cross product terms in $(5)$ are $0$. Then adding $|\mathbf{u}_m|$ I get \begin{align} \int_U\sum_{i,j=1}^n a_t^{ij}(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}\;dx &\leq C\int_U\sum_{i,j=1}^n(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}\;dx \\&= C\int_U\sum_{i=1}^n|(\mathbf{u}_m)_{x_i}|^2+\sum_{i\neq j}^n(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}\;dx\\&=C\int_U\sum_{i=1}^n|(\mathbf{u}_m)_{x_i}|^2\;dx \\ &\leq C\int_U |\mathbf{u}_m|^2+\sum_{i=1}^n|(\mathbf{u}_m)_{x_i}|^2\;dx \\&=C\|\mathbf{u}_m\|_{H^1_0(U)}^2 \end{align}
