Behaviors Of Solutions To Keller-Segel Systems

In this thesis,we study the following three models from Keller-Segel systems:the Keller-Segel system with nonlinear sensitivitythe Keller-Segel-Navier-Stokes system with matrix-valued sensitivityas well as the attraction-repulsion Keller-Segel system Here the smooth bounded domain ????RN,a ?2/N,S ? C²???? ×[0,??²)^N×N,??C^1+????with???0,1?,x,??0,and ?,?,?,?>0.The thesis consists of the following five chapters:In Chapter 1,we introduce the background with rich conclusions on Keller-Segel systems.In Chapter 2,we study the Keller-Segel system with nonlinear sensitivity.Under the homo-geneous Neumann boundary condition ?n·v = ?c·v = 0,we discuss the global boundedness of solutions,where v denotes the unit outer normal of ???.Denote n0=1/|?|??n0?x?dx.We prove that for a ? max{1,N/1},N? 1,the system possesses global classical solutions decaying to the constant steady state?n0,n0?exponentially provided ||u0||Lq*???and ||?v0||Lp*???,small enough with If a ??3/2,1?,N? 3,||n0||L2/Na???and ||?C0||LNa???sufficiently small,then the above result still holds to the corresponding mild solutions of the system.Chapter 3 considers the Keller-Segel-Navier-Stokes system in liquid environments.LetN ?{2,3},|S|?C_s for some C_s>0.Under the boundary condition Vc·v =??n-nS?x,n,c?·?c?·v =0,u = 0,we prove that there exists ?₀>0 such that if ||n0||L2/N???,||?co||LN???,||u0||LN?????0,then the system possesses global classical solutions decaying to?n0,n0,0?exponentially.In Chapter 4,we deal with finite time blow-up on nonradial solutions to the Keller-Segel system of attraction-repulsion model under the the homogeneous Neumann boundary condition?n·v = ?c·v = ?w · v = 0.For N = 2,we establish that the finite time blow-up of nonradial solutions occurs at x0 ?? whenever x?-??>0,the initial mass ?? no?x?dx>8?/?x?-???,and ?? n0?x?|x-x0|²dx sufficiently small.Chapter 5 summarizes the main results of the thesis,and proposes further problems to be studied.