Qualitative Analysis Of A Parabolic-elliptic Repulsion Chemotaxis Model With Two Species

We consider the parabolic-elliptic repulsion chemotaxis modelin a bounded domain ?(?)Rn(n?2)with smooth boundary,where u=u(x,t)and v =v(x,t)denote the cell density of the two species respectively,w = w(x,t)represents the concentration of a signal,d1>0,?1>0,d2>0,X2>0 and ?>0 are parameters.We study the local solvability,the global solvability,boundedness,the existence of nontrivial stationary solutions and asymptotic behavior of the solutions as t?+?.First,based on a method of semigroup,it is proved that the model has a unique local solution.Then,it is further proved that the model has a unique globally-in-time bounded solution via the LP-estimate technique and Moser's iteration method.Finally,by the Lyapunov functional approach,it is shown that the solution converges to a nontrivial stationary solution exponentially in L?(?)as t?+?.