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

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