open access

https://hdl.handle.net/2115/15868

An instability criterion for activator–inhibitor systems in a two-dimensional ball

英語

Nodal domain

2006-10-15

Elsevier Inc.

Journal of Differential Equations

229(2)

494-508

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.

学術雑誌論文

Accepted Manuscript

0022-0396

1090-2732