I am trying to solve both of these exercises:
(Problem 1) Let $u(x,t)$ be a solution of
$$\begin{cases}u_{tt}=c^2u_{xx}+ku,\ \ x\in\mathbb{R},\ \ t>0\\ u(x,0)=u_t(x,0)=0,\ \ x\in\mathbb{R} \end{cases}$$
in which $c>0$ and $k\geq 0$. Take $(x_0,t_0)$, with $x_0\in\mathbb{R}$ and $t_0>0$. Define
$$W(t) = \int_{x_0-c(t_0-t)}^{x_0+c(t_0-t)} u^2_t+c^2u_x^2+ku^2 dx $$
Show that $W(0)=0$, $W(t)\geq 0$, and $W'(t)\leq 0$, for $0<t<t_0$. Conclude that $W(t)=0$, for $0<t<t_0$.
(Problem 2) Show that the problem
$$\begin{cases}u_{tt}=c^2u_{xx}+ku,\ \ x\in\mathbb{R},\ \ t>0\\ u(x,0)=f(x),\ \ x\in\mathbb{R}\\ u_t(x,0)=g(x),\ \ x\in\mathbb{R} \end{cases}$$
in which $K\geq 0$, $f\in C^2(\mathbb{R})$, $g\in C^1(\mathbb{R})$, has at most one solution.
What I accomplished so far:
(Problem 1) I was able to show that $W(0)=0$ and $W(t)\geq 0$, for $0<t<t_0$. To show that $W'(t)\leq 0$ in that same interval I know that it is sufficient to show that $W$ is monotone not increasing in there, so I was trying to proof that for $t_1,t_2\in (0,t_0)$ with $t_1\leq t_2$, then $W(t_1)\geq W(t_2)$, but I was not able to get anywhere from there, I tried using the definition of $W$ and the fact that $u_t,u_x$ and $u$ are continuous (since $u$ is solution of the PDE), can anyone help me from here?
(Problem 2) Here I thought of using the fact that if we have $u$ and $v$ two solutions of the problem in hand, then $F = u-v$ is a solution of the PDE presented in Problem 1, then the problem reduces to showing that that PDE has a unique solution, and in that case it has to be the trivial solution $F=0$ (since this is a solution indeed). My biggest problem is how to use the conclusion of Problem 1 in order to show that the solution must be unique.