Abstract
In this paper, we deal with the following quasilinear attraction–repulsion model: \t\t\t{ut=∇⋅(D(u)∇u)−∇⋅(S(u)χ(v)∇v)+∇⋅(F(u)ξ(w)∇w)+f(u),x∈Ω,t>0,vt=Δv+βu−αv,x∈Ω,t>0,0=Δw+γu−δw,x∈Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω\\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}$$ \\textstyle\\begin{cases} u_{t}=\\nabla \\cdot (D(u)\\nabla u)-\\nabla \\cdot ( S(u)\\chi (v)\\nabla v)+ \\nabla \\cdot ( F(u)\\xi (w)\\nabla w)+f(u), &x\\in \\varOmega , t>0, \\\\ v_{t}=\\Delta v+\\beta u-\\alpha v, &x\\in \\varOmega ,t>0, \\\\ 0=\\Delta w+\\gamma u-\\delta w, &x\\in \\varOmega , t>0, \\\\ u(x,0)=u_{0}(x), \\quad\\quad v(x,0)= v_{0}(x), &x\\in \\varOmega \\end{cases} $$\\end{document} with homogeneous Neumann boundary conditions in a smooth bounded domain varOmega subset R^{n} (ngeq 2). Let the chemotactic sensitivity chi (v) be a positive constant, and let the chemotactic sensitivity xi (w) be a nonlinear function. Under some assumptions, we prove that the system has a unique globally bounded classical solution.
Highlights
Where Ω ⊂ Rn (n ≥ 2) is a bounded domain with smooth boundary, and ∂∂ν denotes the derivative with respect to the outer normal of ∂Ω, α, β, γ, and δ are positive parameters, and χ(v) and ξ (w) represent chemosensitivity
For the case χ (v) = χv with a positive constant χ < n2, a global classical solution is explored by Winkler [18]
In the case where χ(v) and ξ (w) are positive parameters in (1.7), D(u) satisfies (1.3), and f (u) satisfies (1.4), a unique global bounded classical solution was deduced by Wang [15]
Summary
We consider a quasilinear attraction–repulsion chemotaxis system with nonlinear sensitivity and logistic source. For the case χ (v) = χv with a positive constant χ < n2 , a global classical solution is explored by Winkler [18]. In the case where χ(v) and ξ (w) are positive parameters in (1.7), D(u) satisfies (1.3), and f (u) satisfies (1.4), a unique global bounded classical solution was deduced by Wang [15]. F (u) satisfies (1.4), the global classical solutions are asserted by Wu and Wu [19], who obtained an important estimate of Ω |∇v|2 dx. Note that this method is not applicable for the general f (u) in our paper. (iii) If m > max{1, nσ+n+2–22σ , nσn–2 }, (1.1) admits a bounded global classical solution. Suppose that there exists a constant C1 such that u Lk(Ω) ≤ C1, t ∈ (0, T )
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