Global solvability and boundedness to a attraction–repulsion model with logistic source

Abstract

In this paper, we deal with an attraction–repulsion model with a logistic source as follows: {ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+μuq(1−u)in Q,vt=Δv−α1v+β1uin Q,wt=Δw−α2w+β2uin Q,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} \ extstyle\\begin{cases} {u_{t}} = \\Delta u - \\chi \ abla \\cdot (u \ abla v) + \\xi \ abla \\cdot (u \ abla w) + \\mu {u^{q}}(1 - u) &\ ext{in } Q , \\\\ {v_{t}} = \\Delta v - {\\alpha _{1}}v + {\\beta _{1}}u &\ ext{in } Q , \\\\ {w_{t}} = \\Delta w - {\\alpha _{2}}w + {\\beta _{2}}u & \ ext{in } Q , \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} where Q=Ω×R+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$Q = \\Omega \ imes {\\mathbb{R}^{+} }$\\end{document}, Ω⊂R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Omega \\subset {\\mathbb{R}^{3}}$\\end{document} is a bounded domain. We mainly focus on the influence of logistic damping on the global solvability of this model. In dimension 2, q can be equal to 1 (Math. Methods Appl. Sci. 39(2):289–301, 2016). In dimension 3, we derive that the problem admits a global bounded solution when q>87\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q>\\frac{8}{7}$\\end{document}. In fact, we transfer the difficulty of estimation to the logistic term through iterative methods, thus, compared to the results in (J. Math. Anal. Appl. 2:448 2017; Z. Angew. Math. Phys. 73(2):1–25 2022) in dimension 3, our results do not require any restrictions on the coefficients.