Global weak solutions for an attraction‐repulsion system with nonlinear diffusion

This paper deals with the attraction‐repulsion chemotaxis system with nonlinear diffusion u t =∇·( D ( u )∇ u )−∇·( u χ ( v )∇ v )+∇·( u γ ξ ( w )∇ w ), τ 1 v t =Δ v − α 1 v + β 1 u , τ 2 w t =Δ w − α 2 w + β 2 u , subject to the homogenous Neumann boundary conditions, in a smooth bounded domain Ω ⊂ R n ( n ⩾ 2 ) , where the coefficients α i , β i , and τ i ∈{0,1}( i =1,2) are positive. The function D fulfills D ( u )⩾ C D u m −1 for all u >0 with certain C D >0 and m >1. For the parabolic‐elliptic‐elliptic case in the sense that τ 1 = τ 2 =0 and γ =1, we obtain that for any m > 2 − 2 n and all sufficiently smooth initial data u 0 , the model possesses at least one global weak solution under suitable conditions on the functions χ and ξ . Under the assumption m > γ − 2 n , it is also proved that for the parabolic‐parabolic‐elliptic case in the sense that τ 1 =1, τ 2 =0, and γ ⩾2, the system possesses at least one global weak solution under different assumptions on the functions χ and ξ .