Stability of nonconstant steady-state solutions to a Ginzburg–Landau equation in higher space dimensions

Jimbo, Shuichi; Morita, Yoshiro

doi:10.1016/0362-546x(94)90225-9

Cited by 51 publications

(49 citation statements)

References 14 publications

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“…In the case of a shadow reaction-diffusion system with the Neumann boundary condition in a one-dimensional interval, all the stable steady states are constant or monotone [N94,NPY01]. See [JM94,K05,KY03,L96,Y02a,Y02b,Y02c] for other stability and instability results. However, the shape of the stable steady states of a large class of reaction-diffusion systems in high-dimensional domains seems not to be known very much.…”

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confidence: 99%

On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains

Miyamoto

2007

Quart. Appl. Math.

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Abstract. We continue to study the shape of the stable steady states of the so-called shadow limit of activator-inhibitor systems in two-dimensional domainswhere f and g satisfy the following: g ξ < 0, and there is a function (u, ξ). This class of reaction-diffusion systems includes the FitzHughNagumo system and a special case of the Gierer-Meinhardt system. In the author's previous paper "An instability criterion for activator-inhibitor systems in a two-dimensional ball " (J. Diff. Eq. 229 (2006), 494-508), we obtain a necessary condition about the profile of u on the boundary of the domain for a steady state (u, ξ) to be stable when the domain is a two-dimensional ball. In this paper, we give a necessary condition about the profile of u in the domain when the domain is a two-dimensional ball, annulus or rectangle. Roughly speaking, we show that if (u, ξ) is stable for some τ > 0, then the shape of u is like a boundary one-spike layer even if D u is not small.

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confidence: 99%

On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains

Miyamoto

2007

Quart. Appl. Math.

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“…We summarize some of main results given in the works [7], [8] and [10] in relevence to the existence of stable non-constant solutions to (1). In the work [7] it was revealed that any convex domain H never admits stable non-constant solutions, that is, the constant solutions, $ = aja| = 1, are only stable solutions in all convex domains.…”

Section: Xii2mentioning

confidence: 90%

“…We remark that all the stable solutions constructed in [7], [8] and [10] don't vanish in the domains. Hence we have the natural query: is there a stable solution with zeros?…”

Section: Xii2mentioning

confidence: 93%

“…In the work [7] it was revealed that any convex domain H never admits stable non-constant solutions, that is, the constant solutions, $ = aja| = 1, are only stable solutions in all convex domains. On the other hand we see from [7], [8] and [10] that for a domain H which is topologically "non-trivial" in some sense, there exist stable non-constant solutions if A is sufficiently large. Here "non-trivial" means "not simply connected" provided that n = 2 or 3 (for the precise definition, see §3).…”

Section: Xii2mentioning

confidence: 99%

“…The existence of stable non-constant solutions to (1) was first proved in [7] for a thin annulus domain H(e) = [x € M 2 : 1 < \x\ < 1 +e} Actually, with polar cordinate (r,^), Equation (1) has the following limit eqaution as e -> 0:…”

Section: Stable Non-constant Solutionsmentioning

confidence: 99%

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Stable solutions and their spatial structure of the Ginzburg-Landau equation

Morita¹

1995

Journées Équations Aux Dérivées Partielles

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Local minimizers with vortices to the Ginzburg‐Landau system in three dimensions

Montero

Sternberg

Ziemer

2003

Comm Pure Appl Math

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We construct local minimizers to the Ginzburg-Landau energy in certain threedimensional domains based on the asymptotic connection between the energy and the total length of vortices using the theory of weak Jacobians. Whenever there exists a collection of locally minimal line segments spanning the domain, we can find local minimizers with arbitrarily assigned degrees with respect to each segment.

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Stability of nonconstant steady-state solutions to a Ginzburg–Landau equation in higher space dimensions

Cited by 51 publications

References 14 publications

On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains

On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains

Stable solutions and their spatial structure of the Ginzburg-Landau equation

Local minimizers with vortices to the Ginzburg‐Landau system in three dimensions

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