A Keller-Segel System Involving Mixed Local And Nonlocal Diffusion Operators With Logistic Term

Chemotaxis,as one of the most basic cellular physiological responses,plays an important role in nature.As one of the most popular biological models,chemotactic model has attracted much attention since it was proposed,where Keller-Segel system is the first mathematical model to describe chemotactic properties.The research on local diffusion and non-local diffusion has been relatively mature up to now.In fact,the study of mixed local and nonlocal diffusion will be more challenging in order to better simulate the diffusion phenomenon and conform to the complex environment of nature.Due to the influence of nonlocal diffusion,it is difficult to use the original energy methods to deal with the regularity results of weak solutions,so we need to find new methods.In addition,due to the influence of local diffusion,it is necessary to give a limited range for the exponents of the nonlocal term in the existence and asymptotic behavior of equations.In this thesis,we mainly study the Keller-Segel system with mixed local and nonlocal diffusion operators.We mainly extend the previous work,promoting the original fractional diffusion to the mixed local and nonlocal diffusion,and we systematically study the global existence and asymptotic behavior of the solution of the system.Because there are mixed local and nonlocal diffusion operators,the situation will be relatively complicated in the research process.The research contents are as follows.In Chapter 1,we mainly introduce the background knowledge of the research content in this paper and the general situation of relevant research progress.In Chapter 2,the Keller-Segel model with mixed local and nonlocal diffusion operators is discussed.We first obtain the regularity estimate of the weak solution by using blow-up arguments combined with the classical Liouville type theorem.Then the local existence and uniqueness of classical solutions are proved by applying semigroup theory and the regularity results obtained above.Furthermore,under certain conditions,the global existence and boundedness of classical solutions are deduced with given initial value.Finally,the asymptotic behavior of the global solution is discussed under strictly positive initial conditions.In Chapter 3,the main work of this thesis is summarized,and some directions worthy of further research are also put forward.