We are given $u_{tt} - c^2u_{xx} + ru_t$. To prove only one solution exists, I am taking w = $u_1 - u_2$, assuming they are both solutions to the given wave equation.
So: $u_{tt} - c^2u_{xx} + ru_t$
which becomes: $w_{tt} - c^2w_{xx} + rw_t$
I know we are suppose to then multiply both sides by $w_t$ and integrate to show that E(t) is decreasing, but I am confused when it comes to the integration. Please do NOT solve the entire question, I'm simply seeking help for the integration. Once I understand that part, I will post the rest of the problem and look for opinions on how I reached the final answer.
Thanks!
After multiplying $$w_{tt} - c^2w_{xx} + rw_t=0$$ by $w_t$ you should use the following identities $$w_tw_{tt}=\frac{1}{2}\frac{d}{dt}w_t^2,\quad w_tw_{xx}=(w_tw_x)_x-w_xw_{tx}=(w_tw_x)_x-\frac{1}{2}\frac{d}{dt}w_x^2.$$ Then using integration by parts you get $$\frac{d}{dt}E(t)=-r\int_Rw_t^2dx,$$ where $$E(t)= \frac{1}{2}\int_R\big(w_t^2+c^2w_x^2\big)dx.$$ Note that if your problem is a boundary-initial value problem the value of $w$ will be zero on the boundary and hence the value of $w$ is zero on the boundary (means integral on the boundary vanishes). If it is just an IVP (means problem defined on whole of $R$) then the boundary terms vanishes too by the finite speed of propagation property of the wave equation. You didn't state but $r$ is probably a positive constant (so the equation is a damping wave eqn.). In this case the integral on the RHS becomes negative and hence we have $0\leq E(t)\leq E(0)=0$. I think you can proceed from this point.