The lifespan for nonlinear wave equation outside of star-shaped obstacle in three space dimensions

In this paper, we consider the initial-boundary value problem of quasi-linear wave equation outside of star-shaped obstacle in three space dimensions. We study the general case that the nonlinear term may explicitly depend on the unknown function itself. By some new weighted space-time L-2 estimates of the linear wave equation for the unknown function itself, we obtained sharp lower bounds of the lifespan T-epsilon >= C/epsilon(2) for the quasi-linear initial boundary value problem.