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Lifespan of Classical Solutions to Quasilinear Wave Equations Outside of a Star-Shaped Obstacle in Four Space Dimensions (1503.02113v1)
Published 7 Mar 2015 in math.AP
Abstract: 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 $T_{\varepsilon}\geq \exp{(\frac{c}{\varepsilon2})}$, which coincides with the sharp lower bound of lifespan estimate for the corresponding Cauchy problem.