  For a system that is governed by the isothermal Euler equations with friction
for ideal gas, the corresponding field of characteristic curves is determined
by the velocity of the flow. This velocity is determined by a second-order
quasilinear hyperbolic equation. For the corresponding initial-boundary value
problem with Neumann-boundary feedback, we consider non-stationary solutions
locally around a stationary state on a finite time interval and discuss the
well-posedness of this kind of problem. We introduce a strict $H^2$-Lyapunov
function and show that the boundary feedback constant can be chosen such that
the $H^2$-Lyapunov function and hence also the $H^2$-norm of the difference
between the non-stationary and the stationary state decays exponentially with
time.
