The energy method for $u_{tt}-du_t-u_{xx}=0, (0,1)\times(0,T) $ It is a problem in Evan's PDE. I want to prove the smooth solution of the following PDE is zero:
$$u_{tt}-du_t-u_{xx}=0, (0,1)\times(0,T) $$$u|_{x=0}=u|_{x=1}=u|_{t=0}=u_t|_{t=0}=0$.

The hint is to use the energy $\frac{1}{2}(||\partial_tu||_{L^2[0,1]}+||\partial_xu||_{L^2[0,1]})$, but when I differentiate the energy, I can only get the first and third term of PDE

When $d>0$ the uniqueness still follows. For the energy $E(t):=\frac12( \|u_t\|_{L^2}^2+\|u_x\|_{L^2}^2)$, we get $$E'(t)= d\int_0^1(u_t)^2~dx\leq d\left(\int_0^1(u_t)^2+(u_x)^2~dx\right)=2d E(t).$$
Then multiplying $e^{-2dt}$ on both sides, we find $$\frac{d}{dt}\left(e^{-2dt}E(t)\right)\leq 0.$$ That means $e^{-2dt}E(t)$ is decreasing in time, but $E(0)=0$ forces $e^{-2dt}E(t)=0$ for $0<t<T$.
