  In this paper we study a semilinear wave equation with nonlinear,
time-dependent damping in one space dimension. For this problem, we prove a
well-posedness result in $W^{1,\infty}$ in the space-time domain $(0,1)\times
[0,+\infty)$. Then we address the problem of the time-asymptotic stability of
the zero solution and show that, under appropriate conditions, the solution
decays to zero at an exponential rate in the space $W^{1,\infty}$. The proofs
are based on the analysis of the corresponding semilinear system for the first
order derivatives, for which we show a contractive property of the invariant
domain.
