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- Hiroki Yagishita

# Professor Hiroki Yagishita

Area and Subject Taught | Nonlinear Analysis |
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Research Theme(s) | Qualitative Theory of Nonlinear Diffusion Equations |

Academic Degrees | Doctor of Mathematical Sciences, University of Tokyo |

Keywords for Research Field | Applied Analysis, Dynamical Systems,Parabolic Partial Differential Equations |

Office Phone Number | 81-75-705-1610 |

## Research Overview

My primary research interest is nonlinear diffusion equations-partial differential equations describing various diffusion phenomena. This field is not just mathematically interesting; these equations are widely used as models in physics, biology and many other areas of natural science and engineering. Our research approach is distinctive in that we investigate global properties of the solutions of initial value problems for parabolic partial differential equations by combining dynamic systems perspectives and techniques (e.g., Lyapunov functions, invariant manifolds and the comparison principle) with conventional techniques based on classical analysis. The following are two basic topics I have actually worked on so far:1. Bistable reaction-diffusion equations have two spatially uniform stable states, and describe phenomena where those states interact through diffusion. A spatially localized structure called a "boundary," which links the two uniform stable states, appears in these equations, and we are studying the motion of these boundaries.

2. If the boundary in bistable reaction-diffusion equations is viewed in coarse grain, it becomes a surface in space, and the rules governing its time evolution become an equation where the speed of movement in the direction normal to the surface is given by the curvature. We are analyzing the time evolution equations for this surface.

## Notable Publications and Works in the Last Three Years

- T. Ushijima and H. Yagisita, Convergence of a three-dimensional crystalline motion to Gauss curvature flow, Japan Journal of Industrial and Applied Mathematics, 22,443-459, 2005.
- H. Yagisita, Blow-up profile of a solution for a nonlinear heat equation wit