Studies On Qualitative Analysis Of Solutions For Several Classes Of Chemotaxis Models And Related Problem

This thesis studies the qualitative analysis of the solutions for several kinds of chemotactic models and their application in the optimal control,including the well-posedness of the solutions and the large time asymptotic behavior,the existence of the optimal control and the maximum principle for optimal control.The thesis consists of six chapters.Chapter one summarizes background and the source of chemotactic model,and then the current situation and development trend of chemotactic model.We also state the main results of this thesis.Chapter two studies the Neumann problem of a quasilinear chemotaxis model with logistic source.We mainly give the relations between the nonlinear diffusion,nonlinear sensitivity coefficient and the logistic term to ensure the well-posedness of the solution.By L~p-estimate techniques and Moser-Alikakos iteration,we show that the system possesses at least one global and bounded weak solution.These results generalize and improve some previously known ones.Chapter three discusses the dynamics of attraction-repulsion chemotaxis models with logistic source.We consider the global existence and large time behaviors of the solutions for several kinds of attraction-repulsion chemotaxis model.We mainly consider that in the case of linear diffusion,there is some relationship between the logistic source and the nonlinear secretion function to ensure the existence of globally bounded classical solutions.These results generalize and improve some previously known ones.Chapter four consider a two-species chemotaxis-competition system with two signals.We using L~ptechnology to prove that the system possesses at least one glob-al and bounded weak solution.at the same time,we also give sufficient conditions for two competing populations to coexistence and for one population to become ex-tinction.Finally,as an application,we consider the optimal control problem and prove the existence of the optimal control.These results generalize and improve some previously known ones.Chapter five deals with a quasilinear chemotaxis-haptotaxis model.We consider the global existence and boundedness of the solution under the condition that the system contains both nonlinear diffusion and nonlinear sensitivity function.These results generalize and improve some previously known ones.Chapter six studies a virus infection model with saturated chemotaxis.First of all,under appropriate conditions,we prove that the system admits at least one global very weak solution.Then,we prove the existence of the optimal control by using the compactness principle.Finally,the maximum principle is obtained by taking the limit of the extreme condition of the approximate optimal control variable in a reasonable sense.These results generalize and improve some previously known ones.