Studies on the sufficient conditions for the global existence of solutions to the exterior problems for nonlinear hyperbolic equations

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We consider the exterior problem for the linear wave equations, and obtained the convergence rate of the solution to its asymptoics. As an application, we also obtained the precise lower bound estimate of the lifespan of the solutions to the exterior problem for semilinear wave equations. We investigated the asymptotic behavior of global solutions to the systems of nonlinear wave equations with multiple propagation speeds. In connection to the exterior problem, we also study the Cauchy problem in the whole space for various kinds of nonlinear systems, such as systems of wave or Klein-Gordon equations and coupled systems of these equations, and investigated the condition for the global existence of solutions and their asymptotic behavior.