一些趋化-扩散-增长模型斑图的定量刻画和解的整体性态

This project is concerned with the blow-up or global existence of time-varying solutions, the existence of non-constant steady states and the nonlinear dynamical behavior for some chemotaxis-diffusion-growth models. These models are quasilinear and non-uniformly parabolic equations with strong-coupling characteristics. In recent years, the well-posedness and dynamical behavior to these models have become important research subjects of partial differential equations and nonlinear dynamics. In this project, we will mainly focus on the following two aspects: (1) The pattern dynamics of chemotaxis-diffusion-growth models. Based on precise linear analyses, higher-order energy estimates and a bootstrap argument developed by Guo and Strauss, we will try to prove that the nonlinear evolution of general initial perturbations for an unstable uniform equilibrium is determined by the fastest-growing mode for the corresponding linearized equation. The results are expected to provide a quantitative description of the spatial pattern formation. (2)Boundedness vs. blow-up of time-varying solutions for chemotaxis-diffusion- growth models. By using the entropy inequality or other types of comparison inequalities, we investigate the blow-up criterion and the corresponding asymptotic behavior of the solutions. The research results in this project can deepen the understanding of dynamic characteristics of pattern formation and provide theoretic foundation for biology experiment and numerical computation.

本项目研究一些趋化-扩散-增长模型时变解的整体存在性与爆破问题，非常数平衡解的存在性和非线性动力学性态. 这些模型是强耦合的拟线性非一致抛物型方程组. 近年来，这些模型的适定性问题和动力学性态是偏微分方程理论和非线性动力学中的重要研究课题.本项目的第一个主要目的是研究趋化-扩散-增长模型的斑图动力学，基于精细的线性化分析、由Guo, Strauss发展的bootstrap技巧和高阶能量估计，我们将证明不稳定一致平衡态的一般初始扰动的非线性演化是由相应的线性化方程的最快增长模式所控制的，期望所得结果能够定量刻画斑图的生成模式. 第二个主要目的是研究趋化-扩散-增长模型时变解的有界性与爆破问题，应用熵不等式或其它比较不等式探讨解的爆破条件及其渐近性质. 通过本项目的研究，将使我们对一些趋化-扩散-增长系统的斑图动力学特性有更深刻的理解，为生物实验和数值计算提供可靠的理论依据.

本项目主要工作如下：(1) 对于带双稳源的单种群体积填充型Keller-Segel趋化模型和带Lotka-Volterra竞争项的经典Keller-Segel模型，我们首先证明无趋化的半线性反应扩散模型不会产生Turing斑图，然后给出趋化导致平衡点不稳定的条件，应用能量估计和bootstrap技巧证明，对于任意初始摄动，它的非线性演化是由相应的线性系统控制的，该结果是对趋化-扩散-增长模型斑图早期生成的一种定量刻画,且隐含着线性化稳定的平衡点是非线性不稳定的。(2) 对于带Lotka-Volterra互惠(或竞争)源的多种群(或两种群) Keller-Segel模型，在适当条件下研究了时变解的整体存在性和平衡点的渐近稳定性。(3) 对于n-种群互惠扩散模型，当退缩扩散项不同时，反应函数的不同反应率可导致它的解爆破或者整体存在且渐近稳定。 (4) 对于一些半线性反应扩散生物模型，研究了时间斑图、空间斑图和时空斑图的存在性和稳定性。