Global existence and boundedness to a two-species chemotaxis-competition model with singular sensitivity

In the present study, we investigate the chemotaxis-consumption system of two competing species which are attracted by the same signal substance {[ u_t=Δ u-χ _1∇· (u/w∇ w)+u(a_1-b_1u-c_1v), x∈Ω , t>0,; v_t=Δ v-χ _2∇· (v/w∇ w)+v(a_2-b_2v-c_2u), x∈Ω , t>0,; w_t=Δ w-(α u+β v)w, x∈Ω , t>0, ]. associated with homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂ R^n(n≥ 1) , where the parameters α , β , χ _i , a_i , b_i , c_i , i=1, 2 are supposed to be positive. When n=1 , it is shown that whenever the initial data (u_0, v_0, w_0) are positive and suitably regular, the associated initial-boundary value problem admits a globally defined bounded classical solution for any χ _i , b_i>0 (i =1,2) . When n=2 , we establish that if max{χ _1, χ _2}<1 , then the global solution exists regardless of the sizes of b_1>0 and b_2>0 , or if min{χ _1, χ _2}≥ 1 , then there are b^*_i(χ _i) (i =1,2)>0 such that the global classical solution also exists when b_i>b^*_i(χ _i) (i =1,2) . Moreover, the global boundedness of the classical solution is determined as well, that is, there exist λ _i(Ω )>0 and γ _i(Ω )>0 such that the global solution ( u , v , w ) is uniformly bounded in time provided that b_i>λ _i(Ω )a_i+γ _i(Ω ) for max{χ _1, χ _2}<1 or b_i>{b^*_i(χ _i), λ _i(Ω )a_i+γ _i(Ω )} for min{χ _1, χ _2}≥ 1 with i=1,2 , respectively. Furthermore, when n≥ 3 , the corresponding initial-boundary value problem possesses a unique global classical solution under the conditions that max{χ _1, χ _2}<√(2/n) and min{b_1/3α +β, b_2/α +3β}>n-2/4n .