Local smoothing estimates for hyperbolic and dispersive equations and applications

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Abstract Strichartz estimates have been obtained for wave equations outside obstacles. As an application, the global (in time) existence of small solutions to semilinear wave equations with power-type nonlinear terms has been proved in 3-D domain outside obstacles.Weighted space-time L^2 estimates have been also obtained for wave equations with space-time dependent variable coefficients. As an application, some new results on local (in time) wellposedness have been proved for quasilinear wave equations with low regularity radial data.