The lifespan for quasilinear wave equations with multiple propagation speeds in four space dimensions

Abstract

We study the Cauchy problem for systems ofquasilinear wave equations with multiple propagation speeds in fourspace dimensions. The nonlinear term in this problem may explicitly depends on the unknown function itself.By some new $L^{\infty}_{t}L^2_{x}$ estimates of theunknown function, combining with some Klainerman--Sideris typeweighted estimates, we get the sharp lower bound of lifespan $T_{\varepsilon}\geq \exp{(\frac{c}{\varepsilon^2})}$ for the quasilinear system.