Global Boundedness Of Solutions To Some Fully Parabolic Keller-Segel Models

The original chemotaxis systems involved in this thesis were introduced by Keller and Segel in 1970.The corresponding models describe some biological processes,where the cells move towards the high concentration gradient of the chemical substance produced by the cells themselves.This results in a kind of interesting cross-diffusion.The studies in this field have been greatly developed since then.In this thesis,we study the following three models from Keller-Segel systems:(?)The full parabolic Keller-Segel system with logistic-type source and nonlinear productionsubject to the homogeneous Neumann boundary conditions with a bounded and smooth domain?(?)RN(N?1),the sensitivity constant ??0,functions f,g?C1(R)satisfy f(s)?s-?u?for s?0,?>0,a>1,and f(0)?0,0 ? g(s)?s(s+1)?f-1 for s?0,with ?>0.(?)The chemotaxis-Stokes system with slow p-Laplacian diffusion and rotationin a bounded and smooth domain ?(?)R3,p>2,subject to the Neumann-Neumann-Dirichlet boundary conditions,wher ??C2(?),f?C1(?×[0,?);R3)?L?(?×(0,?);R 3),|S(x,n,c)| ?S0(c)(1+n)-?.(?)The fully parabolic chemotaxis system with p-Laplacian diffusion and logistic-type sourcesubject to the non-fiux boundary conditions with a smooth and bounded domain ?(?)RN(N?1),p>1,S? C2([0,?)),S(0)=0,0?S(s)?b0(s+1)? for s?0,f:R?R,f(0)?0,f(s)?b-?sr for s?0,b?0,?>0 and r?1.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 obtain that the model(?)admits globally bounded classical solutions whenever ?<?-1,or ?=?-1 with ? large enough,or ??(0,2/N).Chapter 3 proves for the system(?)that if a+4/3p>25/9 and 11p+6a+2ap>23,then there exist globally bounded weak solutions.Chapter 4 obtains for the system(?)that if p>N?/N+1+N(N+q*)/(N+1)(N+2q*)+1 with q*=max{N/(N-2)+,Nr/(N+2-2r)+},then the problem possesses globally bounded weak solutions.Chapter 5 summarizes the main results of the thesis,and proposes further problems to be studied.