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

Abstract

This paper deals with the attraction‐repulsion chemotaxis system with nonlinear diffusion ut=∇·(D(u)∇u)−∇·(uχ(v)∇v)+∇·(uγξ(w)∇w), τ1vt=Δv−α1v+β1u, τ2wt=Δw−α2w+β2u, subject to the homogenous Neumann boundary conditions, in a smooth bounded domain , where the coefficients αi, βi, and τi∈{0,1}(i=1,2) are positive. The function D fulfills D(u)⩾CDum−1 for all u>0 with certain CD>0 and m>1. For the parabolic‐elliptic‐elliptic case in the sense that τ1=τ2=0 and γ=1, we obtain that for any and all sufficiently smooth initial data u0, the model possesses at least one global weak solution under suitable conditions on the functions χ and ξ. Under the assumption , it is also proved that for the parabolic‐parabolic‐elliptic case in the sense that τ1=1, τ2=0, and γ⩾2, the system possesses at least one global weak solution under different assumptions on the functions χ and ξ.