Global Existence And Asymptotic Behavior Of Solutions To The Keller-Segel Model In Mathematical Biology

This thesis is devoted to the study of the solutions of the Keller-Segel model came from biology, the most important feature of Keller-Segel model is chemotaxis, so Keller-Segel model is sometimes called the chemotaxis model. Chemotaxis is the directed movement of the microorganism or the cell as a response to gradients of the concentration of the chemical signal substance in their environment, so many phenomena in biology can be explained by Keller-Segel model. We main consider global existence and asymptotic behavior of solutions on Keller-Segel model.The thesis consists of six parts.In Chapter 1, we mainly give the background and the development of the related topics and introduce the main content of the present thesis.In Chapter 2, we consider the parabolic-elliptic Keller-Segel model with logistic source. When the diffusion exponent greater than 22n- or the diffusion exponent less than 22n- but the logistic damping effect is sufficiently lagre, we obtain the model possesses a global classical solution which is uniformly bounded. Moreover, when the diffusion exponent less than 22n- and the logistic damping effect is rather mild, we proved that the weak solutions are global existence. Finally, it is asserted that the solutions approach constant equilibria in the large time for a specific case of the logistic source.In Chapter 3, the parabolic-parabolic Keller-Segel model with logistic source is considered. For the case of liner chemotaxis model, we obtain the model possesses a global classical solution which is uniformly bounded by weight function; for the case of nonliner chemotaxis model, we give the inteply of diffusion function and chemsty function which obtain the solutions are global existence and global boundedness.In Chapter 4, we consider the Keller-Segel model with consumption of chemoattractant, where the chemical signal substance may be consumed by the cells themselves, but not produced. For example, bacteria often swim toward higher concentration of oxygen to survive. It is proved that the corresponding initial-boundary value problem possesses a unique global classical solution that is uniformly bounded provided that the diffusion exponent greater than22n-. Moreover, if62n4-+, wheren 33, we only obtain global solutions by lyapunov function.In Chapter 5, we consider the Keller-Segel model with consumption of chemoattractant and logistic source. In the absence of the logistic source for the model, by improve the Gargliardo-Nirenberg inequlity, the model possesses a global classical solution that is uniformly bounded provided that some technical conditions are fulfilled. Moreover, using Maximal Sobolev Regularit, it is shown that the corresponding initial-boundary value problem possesses global solution under the interplay of the nonlinear diffusivity, nonlinear chemosensitivity and logistic source. For the liner chemotaxis model, we also proved the model has global and bounded solution if 0()|| ||Lv??is sufficiently small.In Chapter 6, we summarize the main results of the thesis, and proposes some interesting problems to be done.