Abstract
This paper is concerned with a chemotaxis system with singular sensitivity and logistic source, \t\t\t{ut=Δu−χ1∇⋅(uw∇w)+μ1u−μ1uα,x∈Ω,t>0,υt=Δv−χ2∇⋅(υw∇w)+μ2υ−μ2υβ,x∈Ω,t>0,wt=Δw−(u+υ)w,x∈Ω,t>0,\\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} $$\\begin{aligned} \\textstyle\\begin{cases} u_{t}=\\Delta u-\\chi _{1}\\nabla \\cdot (\\frac{u}{w}\\nabla w)+\\mu _{1}u-\\mu _{1}u^{\\alpha }, &x\\in \\varOmega , t>0, \\\\ \\upsilon _{t}=\\Delta v-\\chi _{2}\\nabla \\cdot ( \\frac{\\upsilon }{w}\\nabla w)+\\mu _{2}\\upsilon -\\mu _{2}\\upsilon ^{\\beta } , &x\\in \\varOmega , t>0, \\\\ w_{t}=\\Delta w-(u+\\upsilon )w, &x\\in \\varOmega , t>0, \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} under the homogeneous Neumann boundary conditions and for widely arbitrary positive initial data in a bounded domain varOmega subset mathbb{R}^{n} (ngeq 1) with smooth boundary, where chi _{i}, mu _{i}>0(i=1, 2) and α, beta >1. It is proved that there exists a global classical solution if max {chi _{1}, chi _{2}}<sqrt{frac{2}{n}}, min {mu _{1}, mu _{2}}>frac{n-2}{n}, alpha =beta =2 for ngeq 2 or any chi _{i}>0(i=1,2), mu _{i}>0 (i=1,2), α, beta >1 for n=1.
Highlights
1 Introduction The chemotaxis system describes a part of the life cycle of cellular slime molds with chemotaxis
Slime molds move towards higher concentration of the chemical substance when they plunge into hunger
When φ(υ) is a constant and f (u) = κu – μu2, [11] established the existence of a global bounded classical solution for suitably large μ and proved that for any μ > 0 there exists a weak solution in the three-dimensional case
Summary
The chemotaxis system describes a part of the life cycle of cellular slime molds with chemotaxis. Before we go to the details of our analysis, let us point out that the global existence, boundedness and stabilization of (weak) solutions to the two-species chemotaxis–fluid system have been established Lemma 2.2 If (1.5) holds, the solution of (1.4) satisfies u(·, t) L1(Ω) ≤ m1, υ(·, t) L1(Ω) ≤ m2, t ∈ [0, Tmax),. In view of the smooth estimates for the Neumann heat semigroup ([26], Lemma 3.1), we obtain c2 satisfying u(·, t) L∞(Ω) ≤ u0 L∞(Ω) + c2χ1 t. ≤ d1 u aL∞(Ω) u 1L–1(aΩ) ∇w Lkp0 ≤ d1 m11–ac u aL∞(Ω), where k = k–k1 , a = 1 – k 1p0 ∈ (0, 1) Inserting this into (3.3), it follows that sup u(·, t) L∞(Ω) ≤ u0 L∞(Ω) + c4 sup u(·, t) aL∞(Ω) + μ1c1T for all T ∈
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