Pattern dynamics and asymptotic analysis in reaction-diffusion systems

在这个项目中，我们用解析和数值相结合的方法研究了以下问题：(1)应用无限维动力系统理论，证明了影子系统稳定解的空间单调性。(2)研究了Gierer和Meinhardt提出的激活剂-抑制剂系统的稳态稳定性。对于环形区域，如果在区域边界具有最大曲率的点处存在局部极大值，则任何定态都是稳定的。(3)通过求解相应的椭圆型边值问题得到反应扩散系统的定态。在这里，我们证明了一些非线性椭圆型方程非平凡解的存在和分支。(4)反应扩散系统中的复杂模式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。我们用渐近方法研究了动力学。(5)形态发生中的激活剂-抑制剂系统、热对流的Swift-Hohenberg方程等可表示为梯度或斜梯度系统。对于这样的系统，我们证明了对于条纹图案，通常可以观察到埃克豪斯不稳定性和之字形不稳定性

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