Local smoothing estimates and applications to nonlinear hyperbolic equations

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The problem of local well-posedness was studied for quasi-linear wave equations with low-regularity radially symmetric data. Emphasis was on the investigation of continuous dependence of solutions on initial data.Combined effects of some two nonlinear terms in the lifespan of small solutions to semilinear wave equations were also studied.Further, the Cauchy problem was studied for systems of quasi-linear wave equations with multiple speeds in two space dimensions. The localized energy estimate for constant-coefficient linear wave equations played a key role in giving an alternative proof of global existence of small solutions.