Global weak solutions for an attraction-repulsion chemotaxis system with p-Laplacian diffusion and logistic source

This paper is concerned with the following attraction-repulsion chemotaxis system with p-Laplacian diffusion and logistic source {ut=∇⋅(|∇u|p−2∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+f(u),x∈Ω,t>0,vt=Δv−βv+αuk1,x∈Ω,t>0,0=Δw−δw+γuk2,x∈Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x),x∈Ω.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\matrix{{{u_t} = \nabla \cdot (|\nabla u{|^{p - 2}}\nabla u) - \chi \nabla \cdot (u\nabla v) + \xi \nabla \cdot (u\nabla w) + f(u),} \hfill & {x \in \Omega ,\,\,t > 0,} \hfill \cr {{v_t} = \Delta v - \beta v + \alpha {u^{{k_1}}},} \hfill & {x \in \Omega ,\,\,t > 0,} \hfill \cr {0 = \Delta w - \delta w + \gamma {u^{{k_2}}},} \hfill & {x \in \Omega ,\,\,t > 0,} \hfill \cr {u(x,0) = {u_0}(x),\,\,\,v(x,0) = {v_0}(x),\,\,\,w(x,0) = {w_0}(x),} \hfill & {x \in \Omega .} \hfill \cr } } \right.$$\end{document}