Problems Related To Chemotaxis Models With Singular Sensitivity

In this thesis,we study problems related to chemotaxis models with singular sensitivity.The global existence,regularity and long time behavior of solutions of three types of chemotaxis model with singular chemotaxis sensitivity are investigated.In chapter 1,we introduce biological background,research progress of classical biological chemotaxis models(such as the Keller-Segel model and chemotaxis(-Navier)Stokes model).Moreover,we introduce the main work and innovations in this thesis as well as preparatory lemmas required for proof.In the second chapter,we consider a chemotaxis-Stokes system with p-Laplacian diffusion and singular sensitivity (?)where Ω(?)R3 is a bounded convex domain with smooth boundary.We discuss the corresponding initial-boundary value problem with Neumann type boundary condition for the first two equations and Dirichlet type boundary condition for the equation of fluid.In this chapter,we discuss the case of p>2,thus p-Laplacian diffusion has the degenerate case,and the chemotaxis term is singular at c=0.In this chapter,we prove the global existence and local boundedness of solutions of this problem for any appropriately regularized initial value,provided that δ>3/2，p≥ 2 and p>min{6δ+1/2(2δ-1),2δ(δ+2)/(2δ+3)(δ-1)}.We mainly employ energy estimate,compact analysis and iterative method in this part.In the third chapter,we consider a Keller-Segel model with gradient dependent chemotactic sensitivity and logistic source (?)where Ω(?)RN(N≥ 2)is a bounded domain with smooth boundary.We investigate the initial-boundary value problem with nonnegative initial data and homogeneous Neumann boundary conditions.Suppose that p,N,α satisfy (?)then we obtain the global existence,eventual smoothness and long time behavior of weak solutions.In this chapter,we mainly employ tools such as Neumann heat semigroup Lp-Lq estimate,Moser iteration,comparison principle etc.In the fourth chapter,we deal with a two-species chemotaxis Navier-Stokes system with singular sensitivity and Lotka-Volterra source(?) where Ω (?)2 is a bounded domain with smooth boundary,and with Neumann type boundary condition for n1,n2,c and Dirichlet type boundary condition for u.In this chapter,we discuss initial-boundary value problem of the above equation and obtain the global existence and uniqueness of classical solution with any appropriately regularized initial value and χ:=max{χ1,χ2}<1.Moreover,we also obtain the global boundedness,asymptotic stability,and asymptotic convergence rate of classical solution,when b1,b2 appropriately large in this chapter.Different from the content of the second chapter,the two species conform to Lotka-Volterra competition relationship and the coupling between equations is stronger.We first establish a basic estimate by establishing a differential inequality of the functional ∫Ωn1pC-qdx+∫Ωn2pc-qdx,and then employ logarithmic transformation,heat semigroup Lp-Lq estimate and iterative methods to prove global existence of the solution.By studying the transformed nonsingular system,a conditional energy functional is constructed,and then the global boundedness of the solution is proved.Furthermore,because the Lotka-Volterra source term has a regularizing effect on the asymptotic behavior of the solution,we derive a result on asymptotic behavior by constructing a Lyapunov functional.