Pattern dynamics and asymptotic analysis in reaction-diffusion systems

反应扩散系统中的模式动力学和渐近分析

• 批准号: 12440023
• 负责人: YANAGIDA Eiji
• 金额: $ 6.46万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440023/
• 关键词: reaction-diffusion system; activator-inthibitor system; skew-gradient systern; bifurcation; pattern formation; eigenvalue analysis; steady state; stability; 興奮一抑制; 自己相似解; 解の爆発; 興奮ー抑制系

In this project, we study the following problems by combining analytical and numerical methods(1) Applying the the theory of infinite dimensional dynamical systems, we show the spatial monetonicity of stable solutions in shadow systems. Also, we obtained a variational characterization of stable steady states for r skew-gradient reaction-diffusion systems(2) We studied the stability of steady states in an activator-inhibitor system proposed by Gierer and Meinhardt. For annular domains, any steady state is stable if it has a local maximum at a point where the boundary of the domain has a maximum curvature(3) Steady states of reaction-diffusion systems are obtained by solving associated elliptic boundary value problems. Here, we showed the existence and bifurcation of non-trivial solution for some nonlinear elliptic equations(4) Complex pattern dynamics obsrved in reaction-diffusion systems can be understood in terms of weak or strong interaction of localized pulses. We studied the dynamics by using asymptotic methods(5) Activator-inhibitor systems in morphogenesis, Swift-Hohenberg equation for thermal convection, etc. can be formulated in terms of gradient or skew-gradient systems. For such systems, we showed that the Eckhaus and zigzag zigzag instabilities can be observed generically for striped Patterns

本项目采用解析和数值相结合的方法研究了以下问题：（1）应用无穷维动力系统理论，证明了影子系统稳定解的空间单调性。（2）研究了Gierer和Meinhardt提出的一类活化剂-抑制剂系统的稳定性。对于环形区域，任何定态是稳定的，如果它在区域边界具有最大曲率的点处有局部极大值。（3）通过求解相应的椭圆边值问题得到反应扩散方程组的定态。本文证明了一类非线性椭圆型方程非平凡解的存在性和分支。（4）反应扩散系统中的复杂斑图动力学可以用局域脉冲的弱相互作用或强相互作用来理解。（5）形态发生中的激活-抑制系统、热对流的Swift-Hohenberg方程等都可以用梯度或斜梯度系统表示。对于这类系统，我们证明了在条纹斑图中可以普遍观察到Eckhaus和zigzag不稳定性

项目成果

E.Yanagida: "Mini-maximizers in reaction-diffusion systems with skew-gradient structure"J. Diff. Eqs. 79. 311-335 (2002)

E.Yanagida：“具有斜梯度结构的反应扩散系统中的最小最大化”J。

J.Nagasawa, I.takagi: "Bifurcating critical points of bending energy under constraints related to the shape of red blood cells"Calculus of Variations.

J.Nagasawa、I.takagi：“在与红细胞形状相关的约束下弯曲能量的分叉临界点”变分微积分。

H.Morishita,E.Yanagida and S.Yotsutani: "Structural change of solutions for a scalar curvature equation"Diff.Int.Eqs.. 14. 273-288 (2000)

H.Morishita、E.Yanagida 和 S.Yotsutani：“标量曲率方程解的结构变化”Diff.Int.Eqs.. 14. 273-288 (2000)

W.-M.Ni, P.Polacik, E.Yanagida: "Monotonicity of stable solutions in shadow systems"Trans. Amer, Math. Soc.. 353. 5057-5069 (2001)

W.-M.Ni、P.Polacik、E.Yanagida：“影子系统中稳定解的单调性”Trans。

S.-I.Ei: "The motion of weakly interacting pulses in reaction-diffusion systems"J.Dyn.Diff.Eqs.. 14(1). 85-137 (2002)

S.-I.Ei：“反应扩散系统中弱相互作用脉冲的运动”J.Dyn.Diff.Eqs.. 14(1)。