Spatio-Temporal Patterns and Finite-Dimensional Dynamical System generated by a Reaction-Diffusion Chemotaxis System

我们研究了Mimura-Tsujikawa系统，由于趋化性平流，整体存在性，爆破和图案形成的解决方案。通过在二维区域中引入一个新的分泌项，证明了解的整体存在性。我们用Jager-Luckhaus方法（Jager-Luckhaus，Trans.Amer.Math.Soc.329（1992））证明了抛物-椭圆近似方程组解的爆破。通过中心流形理论的约化研究了溶液的斑图形成。在所得解中，一个常数解横向分叉为一个不稳定的六边形解，分叉后的分支反转并稳定，同时条纹解引起一个不稳定的音叉分叉.

We examined the Mimura-Tsujikawa system, associated with advection due to chemotaxis, and global existence, blow-up and the pattern formation of solutions. The global existence of solutions was demonstrated by newly introducing a secretion term in a two-dimensional domain. We proved the blow-up of solutions only for the parabolic-elliptic approximated system by using the method of Jager-Luckhaus (Jager-Luckhaus, Trans. Amer. Math. Soc. 329(1992)). The pattern formation of the solutions was investigated by the reduction employing the center manifold theory. In the resulting solution, a constant solution was transversally bifurcated to an unstable hexagonal pattern solution, after which the bifurcated branches were inverted and become stable and simultaneously the stripe pattern solution caused an unstable pitch fork bifurcation.