PDE Seminar (2016/4/8): Reaction-diffusion approximation for understanding pattern formations through non-local interactions, Yoshitaro Tanaka (Hokkaido University)

Date
2016-4-8 16:30 - 2016-4-8 17:30
Place
Faculty of Science Building #3, Room 309
Speaker/Organizer
Yoshitaro Tanaka (Hokkaido University)
 
Recently, various non-local interactions have been observed, for example, in the neural firing phenomena, non-local dispersal and pigment cells in fish skin. To investigate the mechanisms the non-local evolution equations characterized by the convolution with the suitable kernels were proposed.

In this talk, to specify the relationship between the destabilization of the stable homogeneous state by the non-local interaction and the shape of the kernel in the convolution, a non-local evolution equation is analyzed in one dimension. It is shown that the non-local evolution equation can be approximated by a reaction-diffusion system through the singular limit analysis. It clarifies that the non-local interaction induces the diffusion driven instability and specifies the relationship between the destabilization of the solution and the kernel shape. Finally, it is shown that, for any even convolution kernel, there is a reaction-diffusion system which approximates it.