Boundedness in a quasilinear fully parabolic two-species chemotaxis system of higher dimension

Abstract

This paper considers the following coupled chemotaxis system: {ut=∇⋅(ϕ(u)∇u)−χ1∇⋅(u∇w)+μ1u(1−u−a1v),vt=∇⋅(ψ(v)∇v)−χ2∇⋅(v∇w)+μ2v(1−a2u−v),wt=Δw−γw+αu+βv,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} u_{t}=\\nabla\\cdot(\\phi(u)\\nabla u)-\\chi_{1} \\nabla\\cdot(u\\nabla w)+\\mu_{1} u(1-u-a_{1} v), \\\\ v_{t}=\\nabla\\cdot(\\psi(v)\\nabla v)-\\chi_{2} \\nabla\\cdot(v\\nabla w)+\\mu_{2} v(1-a_{2}u-v), \\\\ w_{t}=\\Delta w-\\gamma w+\\alpha u+\\beta v, \\end{cases} $$\\end{document} with homogeneous Neumann boundary conditions in a bounded domain Omegasubsetmathbb{R}^{N} (Nge3) with smooth boundaries, where chi_{1}, chi_{2}, mu_{1}, mu_{2}, a_{1}, a_{2}, α, β and γ are positive. Based on the maximal Sobolev regularity, the existence of a unique global bounded classical solution of the problem is established under the assumption that both mu_{1} and mu_{2} are sufficiently large.