Numerical analysis of attractors determined from nonlinear diffusion systems

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In 2002, we are concerned mainly with constructing stable and reliable discretization scheme for nonlinear parabolic systems. We invited Prof. Favini (Bologna) and Prof. Park (Pusan) who are both the experts in the theory of Abstract Parabolic Evolution Equations. We also contacted with many Japanese researchers to discuss the various subjects concerning to our project. By those activities, we succeeded in constructing a discretization scheme which is globally stable and has a global reliability for exponential attractors determined from nonlinear diffusion systems.In 2003, we are concerned mainly with systematic numerical computations of nonlinear parabolic systems in Physics, Engineering and Biology. We invited Prof. Efendiev (Stuttgart) who is the expert in the theory of Infinite Dimensional Dynamical Systems, and contacted with Prof. Mimura (Hiroshima) who is the presenter of a well-known chemotaxis-growth model. By those activities, we found out that the chemotaixis-growth model admits various pattern solutions, like the network pattern as a pattern With gradual change, the target and perforated target patterns as short range patterns and the honeycomb pattern, the stripe pattern, the perforated pattern and the moving spots pattern as long range patterns. The chemotaxis-growth system has therefore a very remarkable structure of pattern formation.