Infinite dimensional relaxation oscillation in aggregation-growth systems

Shin-Ichiro Ei; Hirofumi Izuhara; Masayasu Mimura

doi:10.3934/dcdsb.2012.17.1859

Article Contents

2012, Volume 17, Issue 6: 1859-1887. Doi: 10.3934/dcdsb.2012.17.1859

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Infinite dimensional relaxation oscillation in aggregation-growth systems

1.
Institute of Mathematics for Industry, Kyusyu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395
2.
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 1-1-1 Higashimita, Tama-ku, Kawasaki, 214-8571

Received: September 30, 2011

Revised: December 31, 2011

Published: August 2012

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Abstract

Two types of aggregation systems with Fisher-KPP growth are proposed. One is described by a normal reaction-diffusion system, and the other is described by a cross-diffusion system. If the growth effect is dominant, a spatially constant equilibrium solution is stable. When the growth effect becomes weaker and the aggregation effect become dominant, the solution is destabilized so that spatially non-constant equilibrium solutions, which exhibit Turing's patterns, appear. When the growth effect weakens further, the spatially non-constant equilibrium solutions are destabilized through Hopf bifurcation, so that oscillatory Turing's patterns appear. Finally, when the growth effect is extremely weak, there appear spatio-temporal periodic solutions exhibiting infinite dimensional relaxation oscillation.

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Mathematics Subject Classification: Primary: 35K57, 35B10; Secondary: 35R15.

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