Research On The Properties Of Solutions To Several Kinds Of Chemotaxis Systems

This paper is devoted to the properties of several kinds of chemotaxis systems,including the existence,uniformly bounded and the asymptotic behavior of global classical solutions.The Chapter 2 is devoted to a parabolic-parabolic chemotaxis system with non-linear diffusion and singular sensitivity:#12 in a bounded domain Ω(?)RN(N≥1),D,S∈ C2([0,+∞))nonnegative,with D(u)=a0(u+1)-α,0≤S(u)≤b0(u+1)β with α<0,β∈R,where the singular sensitivity satisfies 0<φ’(v)≤x/vk.In addition,f:R→R is a smooth function satisfying f(u)≡ 0 or generalizing the logistic source f(u)=ru-μum.It is shown that for the case without a growth source,if 2β-α<2,the corresponding system possesses a globally bounded classical solution.For the case with a logistic source,if N=1 and m>1 or N>2 with m>2β+1,if 2β+α<2,then the corresponding system has a globally classical solution.The Chapter 3 is devoted to a chemotaxis-consumption system with nonlinear diffusion and singular sensitivity:#12 in a smooth bounded domain Ω(?)RN(N≥ 2),with positive parameters χ,∈ and D(u)≥δ(u+ε)m-1,f(u)≤K(u+ε)α,g(u)≤σ(u+ε)β for δ,K,σ,ε,β>0,m≥1 and α∈R.The main goal of this chapter is to prove the existence of global classical solutions when α<N/4,β≤1 and m>α+N/4.The Chapter 4 is devoted to a chemotaxis system with singular sensitivity and logistic source:#12 in a smooth bounded domain Ω(?)RN(N≥2),with χ,μ,k>0,k ∈ R and g(u)(?)uβ.For N≥2,we show that the system admits a global classical solution provide χ<(?),β<2k/N+2 and k>1+2/N.Murthermore,for N=2,we obtain the boundedness and(u,v,|▽v|/v)→(1k/μ)1/k-1,0,0)in L∞(Ω)sense as t →∞ provide k>0,∞<1,k∈N and μsufficiently large.The Chapter 5 is devoted to a chemotaxis-haptotaxis system with nonlinear dif-fusion:#12 in a bounded domain Ω(?)R3 corresponding to zero-flux boundary conditions,with X,ξ,μ,λ>0,k>1,a ∈ R,and D(u)≥ CD(u+1)m-1.It is shown that if α>2-k and β>7/6-2k/3,the corresponding system possesses a classical solution which is globally bounded.The result improved the work proposed in[88],in which,the global boundedness of solutions is established for m>0,α>0,β≥0 and k=2.