  We study the initial-boundary value problem of quasilinear wave equations
outside of a star-shaped obstacle in four space dimensions, in which the
nonlinear term under consideration may explicitly depend on the unknown
function itself. By some new $L^{\infty}_{t}L^{2}_{x}$ and weighted
$L^{2}_{t,x}$ estimates for the unknown function itself, together with energy
estimates and KSS estimates, for the quasilinear obstacle problem we obtain a
lower bound of the lifespan $T_{\varepsilon}\geq
\exp{(\frac{c}{\varepsilon^2})}$, which coincides with the sharp lower bound of
lifespan estimate for the corresponding Cauchy problem.
