People

Email	nishii(@math.sci.osaka-u.ac.jp)
Research	Nonlinear partial differential equations
Keywords	Wave equation, Schrodinger equation, Klein-Gordon equation
URL

My research area is nonlinear partial differential equations. In particular, I am interested in the initial value problem for nonlinear wave equations, nonlinear Klein-Gordon equations and nonlinear Schrodinger equations, which are typical examples of hyperbolic and dispersive equations arising from wave phenomena in a broad sense.

The initial value problem is to find a solution to a differential equation that satisfies given conditions (the initial data) at the initial time. For nonlinear problems, there is generally no explicit formula for solutions. Therefore, we investigate the existence and properties of solutions using mathematical analysis. In this context, the properties of the nonlinear terms play an important role in determining whether solutions exist globally in time and how they behave.

For the nonlinear partial differential equations mentioned above, it is known that global solutions exist for small initial data when the nonlinear terms are of sufficiently high order. Furthermore, such solutions asymptotically behave like solutions to the corresponding linear equations, called free solutions. In contrast, for nonlinearities of critical power, global existence is generally not guaranteed, and even when global solutions exist, they may not behave like free solutions. To understand these phenomena, it is important to investigate how the structure of nonlinear terms influences the properties of solutions.

I study the relationship between the structure of nonlinear terms and the long-time behavior of solutions, with particular interest in nonlinear effects appearing in asymptotic profiles of solutions and their energy decay properties.

Yoshinori NISHII