Pattern dynamics and asymptotic analysis in reaction-diffusion systems

本课题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，证明了阴影系统稳定解的空间货币性。此外，我们得到了r斜梯度反应-扩散系统稳定稳态的变分表征。(2)我们研究了Gierer和Meinhardt提出的活化剂-抑制剂系统的稳态稳定性。对于环形区域，如果在区域边界曲率最大的点处有一个局部极大值，则任何稳态都是稳定的。(3)通过求解相关的椭圆边值问题，得到反应扩散系统的稳态。(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或强相互作用来理解。(5)形态发生中的活化剂-抑制剂系统、热对流中的Swift-Hohenberg方程等可以用梯度系统或斜梯度系统来表示。对于这样的系统，我们证明了埃克豪斯和之字形之字形不稳定性可以在条纹图案中普遍观察到

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