Singular limit system and pattern formation of some reaction-diffusion system

(1)趋化-生长模型和吸收诱导相变模型能够在平流项的反应-扩散系统框架内进行处理。用一般半群理论证明了这些系统局部解的存在性。利用先验估计和比较定理，证明了这些系统在合适的泛函空间中存在全局解。此外，我们还证明了指数吸引子的维数是有限的，指数吸引子是由Temam等人引起的控制系统动力学的某种性质。(2)指数吸引子的维数一般很难确定，除了非系统的反应扩散方程和所考虑的区域的一维空间维数。为了证明这一点，我们考虑了静止解和运动解的存在性及其稳定性。在二维平面上证明了趋化生长模型的平面脉冲平稳解、平面行进锋解和径向对称脉冲平稳解。为了证明这些解的稳定性，我们必须估计系统线性化特征值问题的特征值分布。由于系统的扩散系数很小，所以用奇异极限分析可以解决这一问题。我们首先证明了对于小系数t，决定稳定性的特征值的主导项对应于由奇异极限系统引起的线性微分常方程的系数，这是由原始反应扩散系统的约简得到的。根据这些结果，由于扩散系数很小，我们将得到吸收诱导相变模型的奇异极限系统。

(1) A chemotaxis-growth model and an absorbate-induced phase transition model are able to treat in the framework of the reaction-diffusion system with advection terms. The existence of the local solution of these systems is proved in the general semi-group theory. By using the a priori estimate and the comparison theorem, it is shown that the global solutions of these system exist in the suitable functional space. Moreover, we prove that the dimension of the exponential attractor, which is some kind of property governed the dynamics of the system due to Temam et. al, is finite by showing the squeezing property.(2) It is generally difficult to determine the dimension of the exponential attractor except for the reaction diffusion equation which is not a system, and the one space dimension of the considered domain. For the first step to show it, we consider the existence of stationary solutions and traveling solutions and their stability. We prove these of the planar pulse stationary solutions, planar traveling front solutions and radial symmetric pulse stationary solutions of the chemotaxis-growth model in 2-dimensional plane. Showing the stability of these solutions, we must estimate the distribution of the eigenvalues of the linearized eigenvalue problem of the system. This problem is solved by the singular limit analysis because that the diffusion coefficient of the system is small. We first show that for the small coefficient t he dominant term of the eigenvalues determined the stability is corresponding to the coefficient of the linear differential ordinary equation due to the singular limit system, which is obtained by the reduction of the original reaction diffusion system. From these results, we will have the singular limit system of the absorbate-induced phase transition model because of the smallness of the diffusion coefficient.