Study On Chemotaxis Model With Multi-species And Multi-stimulus

In this thesis,we deal with the global boundedness and finite-time blow-up of solutions to multi-species and multi-stimulus chemotaxis models,such as those of two-species with one-stimulus,or two-stimulus with one-species.This thesis consists of the following five chapters:In Chapter 1,we introduce the background of Keller-Segel systems of chemotaxis models and the current progress of the filed,as well as the main results of this thesis.Chapter 2 deals with two-species quasilinear parabolic-parabolic Keller-Segel model uit=?.(?i(ui)?ui)-?V ·(?i(ui)?v),i = 1,2,vt =?v-v + u1 + u2 in ?×(0,T),subject to the homogeneous Neumann boundary conditions,with bounded domain ?(?)RN,N? 2.We prove that if ?i(ui)/?i(ui)?Ciuiai for ui>1 with 0<ai<N/2 and Ci>0,i = 1,2,then the solutions are globally bounded,while if ?i(ui)/?i(ui)? C1u1a1 for u1>1 with ? = BR,a1>N/2,then for any radial u2O ? C0(?)and m1>0,there exists positive radial initial data u10 with ??u10 = m1 such that the solution blows up in a finite time Tmax in the sense lim,Tmax ||u1(·,t)+ u2(·,t)||L?(?)=?.In particular,if a1>N/2 with 0<a2<N/2,the finite time blow-up for the species u1 is obtained under suitable initial data,a new phenomenon unknown yet even for the semilinear Keller-Segel model of two species.Chapter 3 studies the quasilinear attraction-repulsion chemotaxis model with logistic source ut = ? ·(D(u)?u)-X? ·(?(u)?v)+??·(?(u)?w)+f(u),TVt = ?v + au-?v,??{0,1},0 = ?w + ?u-?w,in bounded domain ?(?)RN,N?1,subject to the homogeneous Neu-mann boundary conditions,D,?,??C2[0,+?)nonnegative,with D(s)?(s + 1)p for s? 0,?(s)?Xsq,?sr? ?sr for s>1,and f ? C?[0,?)satisfying f(s)??s(1-sk)for s>0,f(0)? 0.If the attraction is dominated by one of the other three mechanisms with max{r,k,p +N/2}>q,then the solutions are globally bounded.Under more interesting balance situations,the behavior of solutions depends on the coefficients involved,i.e.,the upper bound coefficientx for the attraction,the lower bound coefficient ? for the repulsion,the logistic source coefficient ?,as well as the constants a and ? describing the capacity of the cells u to produce chemoattractant and chemorepellent respectively.Three balance situations(attraction-repulsion balance,attraction-logistic source balance,and attraction-repulsion-logistic source balance)are considered to establish the boundedness of solutions for the parabolic-elliptic-elliptic case(with?= 0)and the parabolic-parabolic-elliptic case(with-? = 1)respectively.Chapter 4 considers the hyperbolic-elliptic-elliptic model of an attraction-repulsion chemo-taxis model with nonlinear productions and logistic source:ut =-x?·(u?v)+??·(u?w)+?u(1-uk),0 = ?v + auq-?v,0 = ?w + ?ur-?w,in a bounded domain ?(?)RN,N?1,subject to the non-flux boundary condition.We at first establish the local existence of solutions(the so-called strong W1,p-solutions,satisfying the hyperbolic equation weakly and solving the elliptic ones classically)to the model via applying the viscosity vanishing method,and then give criteria on global boundedness versus finite time blow-up for them.It is proved that if the attraction is dominated by the logistic source or the repulsion with max{r,k}>q,the solutions would be globally bounded;otherwise,the finite time blow-up of solutions may occur whenever max{r,k}<q.Under the balance situations with q = r = k,q = r>k or q = k>r,the boundedness or finite-time blow-up would depend on the sizes of the coefficients involved.In Chapter 5,we summarize the main results of the thesis,and propose the problems to be considered.