Lifespan of Classical Solutions to Quasilinear Wave Equations Outside of a Star-Shaped Obstacle in Four Space Dimensions

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 $L_t^{\mathrm{\infty }}{L}_x^2$ and weighted $L_{t,x}^2$ 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.