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

Abstract

This paper is concerned with the following attraction-repulsion chemotaxis system with p-Laplacian diffusion and logistic source

$$\left\{ {\matrix{{{u_t} = \nabla \cdot (|\nabla u{|^{p - 2}}\nabla u) - \chi \nabla \cdot (u\nabla v) + \xi \nabla \cdot (u\nabla w) + f(u),} \hfill & {x \in \Omega ,\,\,t > 0,} \hfill \cr {{v_t} = \Delta v - \beta v + \alpha {u^{{k_1}}},} \hfill & {x \in \Omega ,\,\,t > 0,} \hfill \cr {0 = \Delta w - \delta w + \gamma {u^{{k_2}}},} \hfill & {x \in \Omega ,\,\,t > 0,} \hfill \cr {u(x,0) = {u_0}(x),\,\,\,v(x,0) = {v_0}(x),\,\,\,w(x,0) = {w_0}(x),} \hfill & {x \in \Omega .} \hfill \cr } } \right.$$

The system here is under a homogenous Neumann boundary condition in a bounded domain Ω ⊂ ℝⁿ(n ≥ 2), with χ, ξ, α, β, γ, δ, k₁, k₂ > 0, p ≥ 2. In addition, the function f is smooth and satisfies that f(s) ≤ κ − μs^l for all s ≥ 0, with κ ∈ ℝ, μ > 0, l > 1. It is shown that (i) if $l>\max\{2k_{1},{2k_{1}n\over{2+n}}+{1\over{p-1}}\}$, then system possesses a global bounded weak solution and (ii) if $k_{2}>\max\{2k_{1}-1,{2k_{1}n\over{2+n}}+{2-p\over{p-1}}\}$ with l > 2, then system possesses a global bounded weak solution.