Boundedness Of Solutions In Two High-dimensional Chemotaxis Systems

With the development of the theory of nonlinear partial differential equations,its theoretical achievements are more and more widely used in biology and other fields.In recent years,many mathematical researchers began to use rigorous nonlinear partial differential equation theory and proof methods to explain and predict some complex biological phenomena,which promoted the development and popularization of biology.This paper discusses nonlinear parabolic equations from other applied research fields such as biology and mathematics.In this paper,the global existence and boundedness of global solutions of two kinds of biological chemotactic models with logistic sources are studied.The specific contents of this paper are as follows:The first chapter introduces the background of the chemotaxis model and its development status,and summarizes the main content of this paper.The second chapter introduces the basic concepts of partial differential equations and the necessary Lemma of the proof process are briefly introduced,and some proofs of the Lemma are given.The third chapter,based on the energy method,the global boundedness of the classical solution of the chemotaxis model with special logistic source and Neumann boundary conditions is proved.Firstly,used the boundedness of the function ?(s)= exp[(1+ ?*s)-r]to establish the local prior estimate of u and w,and then used the energy estimation method,it is proved that the corresponding initial-boundary value problem possesses a unique global classical solution that is uniformly bounded.The fourth chapter,the boundedness of the global solution of the chemotaxis model with general logistic source and Neumann boundary conditions is discussed and the relationship between parameters and dimensions in the model is obtained.Firstly,used the energy estimation method and Gagliardo-Nirenberg inequality to establish the local prior estimate of u and w,and then used Moser iteration,this problem admits a unique global classical solution that is uniformly-in-time bounded.