Pattern dynamics and asymptotic analysis in reaction-diffusion systems

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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