- Partial differential equations, in particular evolution equations
- Qualitative behavior of solutions to parabolic equations
- Mathematical biology, in particular chemotaxis and tumor invasion
My interest is in a system of partial differential equations, especially, the so called Keller-Segel system. This system describes a biological phenomenon chemotaxis. From a mathematical view point, the Keller-Segel system has two conflicting effects, diffusion and concentration. A considerable literature has been devoted to analysis of various behaviors of solutions (global existence, finite time blow up, etc). I focus on energy structures of the system and categorize behaviors of solutions by localizing or generalizing the structure. Recently, I am trying to research structures of solutions by regarding the system as a perturbation of a single equation. Moreover, by applying mathematical methods which were developed in the study of the Keller-Segel system, I study mathematical models describing a cancer invasion phenomenon.