Existence And Stability Of Solutions To Three Classes Of Partial Differential Equations Arising From Chemotaxis

In the thesis, we investigate the traveling wave solution to the parabolic-hyperbolic systems, and prove that there is a pair of non-trivial steady state solution to the attractive-repulsive chemotactic model including the logarithm chemosensitivity, and establish the global existence of solution to the attraction-repulsion Keller-Segel model including the fast diffusion and nonlinear source term, and establish the chemotactic model of some coupled parabolic equations arising from Alzheimer’s Disease.In chapter 1, we firstly introduce the derivation of a macroscopic model describing the chemotaxis phenomenon based on the aggregation stage of Dictyosteliumdiscoideum’s life cycle, which was first discovered by Keller and Segel(1970), and hence it is now called Keller-Segel model (K-S model). The original K-S model which can reflect the macroscopic behavior of the cell in-cluding four strongly coupled partial differential equations, and their simplified formulation including two strongly coupled parabolic equations called "Mini-mal Model". Next, we present the derivation of the chemotactic model from the microscopic point of view. Patlak(1953), Othmer and Stevens(1997) estab- lished the microscopic model according to the microscopic criterion of behavior of the cell by applying the random walk and transition probability theories. Moreover, Othmer and Stevens obtained the macroscopic model by taking the limit of microscopic model. Finally, we show some revised K-S model based on the classical K-S model, for example:pde-ode coupled K-S model、volume-filling K-S model、attraction-repulsion K-S model including one cell and two chemicals and chemotaxis-fluid coupled chemotaxis model, and so on.In chapter 2, we show the existence of shock solutions to a hyperbolic-parabolic system arising from repulsive chemotaxis. Here we consider the gen-eral Riemann problem and obtain the shock curves in parameter-characterized forms. We give the shock structure by studying the traveling wave solutions and prove that the traveling wave speed is equal to the shock speed. More-over, we derive an entropy-entropy flux pair to prove the uniqueness of the weak shock solutions.In chapter 3, we study the global existence of solutions to a strongly coupled parabolic-parabolic system of chemotaxis arising from the theory of reinforced random walks. This system is an attraction-repulsion chemotaxis model with fast diffusive term and nonlinear source subject to the Neumann boundary conditions. Such fast diffusion guarantees the global existence of solutions for any given initial value in a bounded domain. Our main results are based on the method of energy estimates, where the key estimates are obtained by a technique originating from Moser’s iterations. Moreover, we notice that the cell density goes to the maximum value when the diffusion coefficient of the cell density tends to infinity.In chapter 4, we mainly consider the instability of homogeneous steady- state solution to the attractive-repulsive Keller-Segel model. To this end, we firstly linearized the system and argued the finite dimensional system in-cluding only one parameter. And then, we obtain a the sufficient condition for destabilization by using the theory of non-negative matrix and graph.In chapter 5, we give some conclusions and discuss some problem for the further study.