Purpose and Background of the Research

Many mathematical models are described by nonlinear partial differential equations and such equations typically have both linear and nonlinear structures therein. The linear part is described by partial differential operators by local space-time variables and the nonlinear part is produced by the interaction between different physical quantities and the linear structure stabilize the system while the nonlinear part causes instability of the model. Between those effects, there exists a sort of problems where the both effects are analytically balanced. We call this type of problem as the “Critical Problems” and it is our main subject of this project. Problems of this type are interesting both from an applied and a theoretical mathematical point of view.

They often lead to new and fascinating open problems.

A serious difficulty in the study of such “critical problems”, is that the analysis derived through perturbation theory is not directly applicable and a new methodology has to be developed.