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An instability criterion for activator–inhibitor systems in a two-dimensional ball
| Title: | An instability criterion for activator–inhibitor systems in a two-dimensional ball |
| Authors: | Miyamoto, Yasuhito Browse this author |
| Keywords: | Instability | | Activator–inhibitor system | | Shadow system | | Reaction–diffusion system | | Nodal curve | | Nodal domain |
| Issue Date: | 15-Oct-2006 |
| Publisher: | Elsevier Inc. |
| Journal Title: | Journal of Differential Equations |
| Volume: | 229 |
| Issue: | 2 |
| Start Page: | 494 |
| End Page: | 508 |
| Publisher DOI: | 10.1016/j.jde.2006.03.015 |
| Abstract: | Let B be a two-dimensional ball with radius R. Let(u(x,y),ξ) be a non-constant steady state of the shadow system ut=Du∆u+f(u,ξ) in B×R+ and τξt =1/|B|∬Bg(u,ξ)dxdy in R+, ∂νu=0 on∂B×R+, where f and g satisfy the following:fξ(u,ξ)<0,gξ(u,ξ) <0 and there is a function k(ξ) such that gu(u,ξ)=k(ξ)fξ(u,ξ). This system includes a special case of the Gierer-Meinhardt system and the FitzHugh-Nagumo system. We show that if Z [Uθ(・)] ≥3, then(u,ξ)is unstable for all τ>0, where U(θ):=u(Rcosθ,Rsinθ) and Z [w(・)] denotes the cardinal number of the zero level set of w(・)∈Cº(R/2πZ). The contrapositive of this result is the following: if(u,ξ)is stable for someτ>0, then Z[Uθ(・)]=2. In the proof of these results, we use a strong continuation property of partial differential operators of second order on the boundary of the domain. |
| Relation: | http://www.sciencedirect.com/science/journal/00220396 |
| Type: | article (author version) |
| URI: | http://hdl.handle.net/2115/15868 |
| Appears in Collections: | 知識メディア・ラボラトリー (Meme Media Laboratory) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)
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Submitter: 宮本 安人
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