Blow Up And Asymptotic Behavior Of The Solutions To Some Chemotaxis KS Systems

Biological mathematics is an interdiscipline which combines mathematics and biology,take biology as its background and research in mathematics methods.In the study of biology mathematics,Keller-Segel model was used primarily to describe the directional movement of biological cells or microorganisms due to the stimulation of certain chemicals in microcosmic state.The directional movement can be called chemotaxis,therefore,the Keller-Segel model is also can be called chemotaxis model.Chemotaxis model has a strong applicability,and can be used to understand wound healing,cellular immune response,the growth of microorganism.This thesis is devoted to investigating some properties of the solutions to some chemotaxis system,such as global existence,large time asymptotic behavior,largr time asymptotic rate,finite time blow up and the blow up time of the solution.The thesis is divided into six chapters as follows:In Chapter 1,we introduce the biological background and the related topics of the Keller-Segel model.Moreover,we will briefly describe the main content of the present thesis.In Chapter 2,we mainly study the parabolic-elliptic chemotaxis Keller-Segel model with logistic source.We find a relationship between the diffusion exponent and the logistic damping effect,that is,when the diffusion exponent is sufficiently large(the cells itself diffuse fast)or the diffusion exponent is small but the logistic damping effect is suitable lagre,we see that the chemotaxis system possesses a global classical solution.Moreover,the solution is uniformly bounded.Besides,for a special logistic source,we investigate the large time asymptotic behavior of the solution to the model.In Chapter 3,we discuss the asymptotic behavior and asymptotic rate of the solution to the parabolic-elliptic chemotaxis Keller-Segel model with logistic source.When the logistic source can be written as u u??-?(?(29)1),and the parameter ? is suitable large,then the solution convergence to a steady states at an exponential rate.In Chapter 4,we study the chemotaxis system with two-species and one chemoattractant under the assumption n(28)2.First,by constructing a function,we show that the solution to the problem will blow up at a finite time if the initial total mass of the two-species is larger than a threshold.Secondly,we find a lower bound of the blow up time.Finally,we prove that when both the initial mass of the two-species lie below some thresholds,respectively,then the solu-tion exists globally and remains bounded.In Chapter 5,we mainly investigate the parabolic-elliptic-elliptic quasilinear chemotaxis attraction-repulsion chemotaxis system with one species cells and two chemical substances which produced by the cells.We first consider that when n(28)3,some assumptions about the diffusion exponent and the logistic damping effect are hold,the global existence of the solution.After that,by constructing a Lyapunov function,we obtain that solution convergence to a steady states at an exponential rate.Finally,when the random motion of cells is ingored,(the movement of cells is affected by chemicals only)we consider the existence of the weak solution.Furthermore,we find a critical threshold such that the weak solution blows up at finite time when the parameter of the logistic source is smaller than the threshold value,however,the weak solution exists globally when the parameter of the logistic source is larger than the threshold value.In Chapter 6,we summarize the work in this paper and put forward the content of the future research.