I would like to know how to solve exercise $10$ of chapter 7 of Evans' Partial Differential Equations, second edition. The problem goes like this:
Suppose $U\subset\mathbb{R}^n$ is an open and bounded set with smooth boundary, and $T>0$. Show there exists at most one smooth solution of this initial/boundary-value problem for the telegraph equation $$\begin{cases} u_{tt}+du_t-u_{xx}=f&\text{ in }(0,1)\times(0,T)\\ u=0&\text{ on }(\{0\}\times[0,T])\cup(\{1\}\times[0,T])\\ u=g,u_t=h&\text{ on }[0,1]\times\{t=0\} \end{cases}$$
I have no idea of what shall I do... if we suppose $u$ and $v$ are regular solutions of the given problem, then $u-v$ is a regular solution to the following problem $$\begin{cases} u_{tt}+du_t-u_{xx}=0&\text{ in }(0,1)\times(0,T)\\ u=0&\text{ on }(\{0\}\times[0,T])\cup(\{1\}\times[0,T])\\ u=0,u_t=0&\text{ on }[0,1]\times\{t=0\} \end{cases}$$ However, I don't know how to prove that any regular solution to the last problem must be identically zero in $(0,1)\times(0,T)$... I think the best way to proceed would be to make some change of variables in order to get rid of the term in $u_t$ above, and get some hyperbollic problem.
I am completely lost, so any help will be appreciated.