Instability of multi-spots in the shadow system for the Gierer-Meinhardt model

Kota Ikeda
Mathematics Institute, Tohuka University

(October 7, 2008 2:00 PM - 3:00 PM)

Instability of multi-spots in the shadow system for the Gierer-Meinhardt model

Abstract

The Gierer-Meinhardt system is a mathematical model describing the process of hydra regeneration. This system has a stationary solution with spotted pattern, and numerical results suggest that such spotted pattern is stable and multi-spots appear. On the other hand, it is shown by numerical simulations that the shadow system, which is a limiting system for the original Gierer-Meinhardt system, has no stable multi-spots. In this talk we study a linearized eigenvalue problem with a nonlocal term and prove the instability of multi-spots in a mathematically rigorous manner under suitable assumptions. In addition, we shall obtain profiles of eigenfunctions corresponding to the unstable eigenvalues.