PDE Seminar (2019/11/22): Dynamics of localized patterns for the equation with nonlocal effects

2019-11-22 16:30 - 2019-11-22 18:00
Faculty of Science Building #3, Room 309
Tsubasa Sukekawa (Hokkaido University), Hiroshi Ishii (Hokkaido University)
• Tsubasa Sukekawa (16:30--17:15)

Standing pulse solutions for linear mass conserved reaction-diffusion system

In this talk, we consider a linear reaction diffusion system with mass conservation. This system come from the model equations for ‘Cell polarity’, which is biological phenomenon, and it is important problem that which pulse-like stationary solution exists or not. We will show that a pulse-like stationary solution exists under some conditions for reaction terms. To prove this, we use the fact the stationary problem is equivalent to a linear integro-differential equation. As time allows, we explain the relation between diffusion coefficients and stationary solution. This talk is based on a joint work with Sungrim Seirin Lee (Hiroshima University), Tomohiro Nakahara (Hiroshima University), Hiroshi Ishii (Hokkaido University) and Shin-Ichiro Ei (Hokkaido University).

• Hiroshi Ishii (17:15--18:00)

Existence of traveling waves to a nonlocal scalar equation with sign-changing kernel

In this talk, we address the existence of traveling wave solutions connecting two constant states to a nonlocal scalar equation with sign-changing kernel. A typical example of such kernel in the neural fields is the Mexican hat type function. We first introduce a new notion of upper-lower-solution for the equation of wave profile for a given wave speed. Then, we construct two different pairs of upper-lower-solutions and use Schauder's fixed point theorem to obtain traveling waves for a continuum of wave speeds under some assumptions. Finally, we analyze wave profiles by showing the limit of the both-hand tails. This talk is based on a joint work with Shin-Ichiro Ei (Hokkaido University), Jong-Shenq Guo (Tamkang University) and Chin-Chin Wu (National Chung Hsing University).