The Study Of A Chemotaxis System With Logistic Source

Chemotaxis is a phenomenon in which chemical signals produced by unevenly distributed substances in space stimulate directional movement of cells or organisms.The chemotaxis model is a partial differential system that describes the chemotaxis of organisms.Keller-Segel model is the most typical representative of chemotaxis model.If the reproduction and death of cells is considered,then there will be logistic source in the mathematical model.Hence,the following chemotaxis system with logistic source will be considered:(?)where ?(?)RN(N? 1)is a bounded domain with smooth boundary,k?R,?>0,?>1,the unknown functionsu(x,t)is the cell density and v(x,t)is the density of the chemoattractant.The global well-posedness of this model in the case of ??2has been established.When the logistic source is quadratic,that is1<?<2,the existence of the global solution of the large initial value has not been studied.Here we prove that when the logistic source is quadratic,the global solution of this problem also exists without relying on the small initial value.This dissertation is divided into the following chapters:In the introduction,we mainly introduce the background of chemotaxis model and some domestic and foreign research status related to this study.In the first chapter,we give the research model and main results as well as some preparatory knowledge.In the second chapter,we prove that when N<6,?>1 or N?6,?>4/3,the initial-boundary value problem has a global weak solution.In the third chapter,we prove that as long as ?>1,the initial-boundary value problem possesses a global generalized solution for any N?1.