$$\def\d{\mathop{}\!\mathrm{d}}$$ In the domain $Q_T = (0,l)\times (0,T)$ we consider the following problem
$$\begin{align} & u_{tt} - a^2 u_{xx} + b(x,t)u_x + b_0(x,t)u_t + c(x,t)u = f(x,t), \tag{$\star$} \\ & u|_{x=0} = 0,\qquad (u_x + ku)|_{x=l} = 0, \notag \\ & u|_{t=0} = \varphi(x), \qquad u_t|_{t=0} = \psi(x), \notag \end{align}$$
where $b(x,t)$, $b_0(x,t)$, $c(x,t)$ and $f$ belong to $C(\overline{Q_T})$.
I need to give an appropriate definition of the energy $E(t)$ to the problem and estimate $E(t)$.
For the case $k\geq 0$, define the energy functional $$ E(t) := \frac12 \int_0^l (u_t^2 + a^2 u_x^2) \d x + \frac12 ka^2 u^2(l,t). $$ Multiplying both sides of the equation $(\star)$ by $u_t$ and integrating from $0$ to $l$ with respect to $x$, we get \begin{align*} \frac{\d E(t)}{\d t} & = - \int_0^l (bu_xu_t + b_0 u_t^2 + cuu_t) \d x + \int_0^l fu_t \d x \\ & \leq C\int_0^l (u_t^2 + u_x^2 + f^2) \d x \\ & \leq C\biggl(E(t) + \int_0^l f^2 \d x\biggr), \end{align*} from which it follows that $$ E(t) \leq E(0)e^{Ct} + C\int_0^t e^{C(t-s)} \int_0^l f^2 \d x\d s. $$
Now the problem is: how to define $E(t)$ for the case $k<0$?