Dynamic Behavior Of Solutions To Keller-Segel Chemotaxis Systems With Singular Sensitivities

In this thesis,we study the following four fully parabolic Keller-Segel chemotaxis system with singular sensitivity under the homogeneous Neumann boundary condition,i.e.,the singular chemotaxis model(?) and the case with logistic source (?) the singular chemotaxis-consumption system (?) and the case with logistic source (?) whee α(?)RN is a smooth bounded domain,χ,α,K,ε,r,μ>0,k>1.The thesis consists of six chapters as follows:Chapter 1 introduces the background and the main conclusions of the four models considered and describes the main results in this thesis.Chapter 2 considers the global existence-boundedness of classical solutions to singular chemotaxis model(1).It is proved that if N≥ 2,κ>0 and χ∈(0,(?)),the system possesses a global classical solution,which is uniformly bounded provided 2 ≤ N ≤8.In particular,by letting κ→∞,this does agree with the known requirement χ∈(0,2/N)obtained for the corresponding parabolic-elliptic model.In Chapter 3,we discuss the influence of logistic source to the solution in the singular chemotaxis model(2).If N = 2,r∈R,χ>0,there exist global classical solutions.Moreover,these solutions are globally bounded if r>χ2/4 for 0<χ≤2,or r>χ-1 for χ>2.In Chapter 4,we deal with the singular chemotaxis-consumption system(3).It is proved that the problem admits a global classical solution if N = 1 and ε,α>0.Moreover,if α,N ≥ 1,then we have for the global classical solution(u,v)that v converses to 0 in the L∞-norm as t →∞with the decay rate established whenever ε∈(ε0,1)with ε0=max{0,1-χ/α||v0||∞(Ω)α-1)}.In Chapter 5,we consider the singular chemotaxis-consumption system(4)with logistic source.It is proved that the problem(4)possesses a global classical solution provided N=1 with k>1,or N ≥2 with k>1 +N/2.In particular,in two dimensional setting,(u,v,|▽v|/v)→((r/μ)1/k-1,0,0)in the L∞(α)sense as t → ∞ provided μ>0 sufficient large.Chapter 6 summarizes the main results in this thesis and proposes the problems to be considered in the future.