Global existence and asymptotic behavior for systems of nonlinear hyperbolic equations

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We studied sufficient conditions for global exitence of solutions and asymptotic behavior of global solutions to the Cauchy problem (or the exterior problem) for (systems of) nonlinear hyperbolic equations. For complex-valued (or systems of) semilinear wave equations in two space dimensions, we obtained sufficient conditions, which are weaker than the so-called null condition, for small data global existence, and also obtained sufficient conditions for the energy decay. For the exterior problem for semilinear or quasi-linear wave equations in two and three spece dimensions, we derived a lower bound for the lifespan of classical solutions when the null condition is not satisfied. We also clarified the null conditions for massless Dirac equations and systems of nonlinear Schroedinger equations, and proved that the global solutions are asymptotically free.