Email katayama(
Nonlinear partial differential equations
Keywords Wave equation, Klein-Gordon equation, Schrödinger equation

My research interest is in nonlinear partial differential equations. To be more specific, I am working on the initial value problem for nonlinear wave equations (in a narrow sense), and also for partial differential equations describing the nonlinear wave propagation in a wider sense, such as Klein-Gordon equations and Schroedinger equations.

The initial value problem is to find a solution to a given partial differential equation with a given state at the initial time (a given initial value). However, in general, it is almost impossible to give explicit expression of solutions to nonlinear equations. Therefore, in the mathematical theory, it is important to investigate the existence of solutions and also their behavior when they exist.

If we consider the initial value problem for the equations mentioned above and if the initial value is sufficiently small, the existence of solutions up to arbitrary time (the existence of global solutions) is mainly determined by the power of the nonlinearity. Especially, when the nonlinearity has the critical power, the existence and non-existence of global solutions depend also on the detailed structure of the nonlinear terms. I am interested in this kind of critical case, and studying sufficient conditions for the existence of global solutions and their asymptotic behavior.