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

我们研究了Mimura-Tsujikawa系统，它与由于趋化性引起的平流、全球存在、爆炸和解的模式形成有关。通过在二维域中引入一个新的分泌项，证明了解的全局存在性。利用Jager-Luckhaus （Jager-Luckhaus, Trans.）的方法证明了仅对抛物-椭圆型近似系统解的爆破性。阿米尔。数学。Soc。329(1992))。利用中心流形理论研究了解的模式形成。在得到的解中，常数解被横向分岔为不稳定的六边形解，分岔后分支倒转并变得稳定，同时条纹解引起不稳定的节叉分岔。

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.