Long-time Behavior Of Solution To A Keller-Segel Chemotactic Model With Indirect Signal In Three-dimensional Case

Chemotaxis refers to the directional movement of insects,cells,bacteria and other organisms stimulated by chemicals in the external environment.Because chemotaxis is widespread in nature,it has attracted the interests of many scholars.In order to describe the dispersal and aggregation behavior of Mountain Pine Beetles(MPB)in forest habitats,Strohm,et al.proposed a model consisting of parabolic-parabolic Keller-Segel equations coupled with ordinary differential equation.On the basis of this model,this paper studies the propagation and aggregation behavior of two different states of MPB.This thesis mainly studies the well-posedness of the simplified three-dimensional Keller-Segel-ODE model with indirect signal:under homogeneous Neumann boundary conditions,where ? ?R3 is a bounded domain with smooth boundary,?>0 is a given parameter,u represents the density of flying MPB in the forest,w denotes the density of nesting MPB,v is the concentration of beetles pheromone and ?um stands for the non-linear diffusion with porous media,respectively.The specific research contents of this thesis are as follows.1.The global existence of bounded weak solution to the initial-boundary value problem is established:Firstly,we obtain some priori estimates of the regularization problem by using the energy estimates,interpolation inequalities and Sobolev embedding inequalities,and then achieve the boundedness of ?u?(·,t)?Lp(?)with p<9(m-1)and m>4/3;Then,the boundedness of ?u?(·,t)?L?(?)is obtained by using the iterative method;Finally,the existence of globally bounded weak solution for m>4/3 is obtained by applying the Aubin-Lions lemma.2.The long-time behavior of the weak solution(u,v,w)for the initial-boundary value problem is established:The convergence of weak solution(u,v,w)is obtained by contradiction,comparison principles of parabolic equation,interpolation inequalities,integral mean-value theorem,etc.,and thus the long-time behavior of weak solution(u,v,w)is obtained.Especially,we prove the weak solution(u,v,w)converges to a constant equilibrium state(u0,0,u0/?).