Research On The Properties Of Solutions To Chemotaxis Systems

There are many nonlinear phenomena,such as chemotaxis and haptotaxis,in the fields of biology,ecology and medicine.Mathematical analysis of these phenomena has become more and more important in understanding these complex processes.The distribution of population density has become one of the interested topics since many nonlinear phenomena are caused by the distribution of population density.The distribution of population density can be modeled by partial differential equations.The study of these partial differential equations with biological background has become one of important issues in the field of partial differential equations.This thesis mainly studies the properties of solutions to the partial differential equations modeling chemotaxis.The main contents and results are as follows:1.We study the initial–boundary value problem of an attraction-repulsion chemotaxis system of parabolic-elliptic type with nonlinear diffusion and logistic source.This system describes the chemotactic movement of cells under the combined effect of chemoattractant,chemorepellent,nonlinear diffusion and logistic source.Firstly,we obtain the local existence and uniqueness of the classical solutions to the system with the nondegenerate diffusion by a fixed point theorem and the parabolic and elliptic regularity.Secondly,using the energy estimates,we show that the system with non-degenerate diffusion admits a unique global bounded classical solution provided that the repulsion prevails over the attraction,or the logistic dampening is sufficiently strong,or the diffusion is sufficiently strong.Thirdly,we show that the system with degenerate diffusion possesses at least one global bounded weak solution under the same assumptions.Finally,we obtain the large-time behavior of the solutions for a specific logistic source.2.We study the initial–boundary value problem of a quasilinear 2D parabolic–parabolic attraction–repulsion chemotaxis system.It is known that the specific semilinear system admits a finite time blow-up solution in a two-dimensional bounded domain when the attraction prevails over the repulsion.We consider the nonlinear diffusion and obtain that both the finite time and infinite time blow-up can be prevented provided that the diffusion is super-linear.Hence,we show that the system admits a global bounded classical solution for the case of non-degenerate diffusion,while the system possesses a global bounded weak solution for the case of degenerate diffusion.3.We study the initial–boundary value problem of a higher-dimensional quasilinear chemotaxis system with consumption of oxygen.In this model the chemical substance such as oxygen is consumed by the cells unlike the two above-mentioned models,where the chemical substance is produced by the cells.Using the L?-bound of the chemical substance,we construct a new interpolation inequality and then establish the combined energy estimates.We show that the system admits a global bounded classical solution for the case of non-degenerate diffusion,while it possesses a global bounded weak solution for the case of degenerate diffusion.4.We study the initial–boundary value problem of a degenerate quasilinear chemotaxis system with rotation.In this system,the chemical substance is consumed by the cells and the chemotactic sensitivity is a tensor.The norm of the tensor has a superlinear growth of the cells' density.Firstly,we construct an approximate problem with non-degenerate diffusion and good boundary conditions.Secondly,we show that the approximate problem admits a global bounded classical solution by the energy estimates.Finally,we obtain that the original system possesses a global bounded weak solution by convergence analysis.