Boundedness in a two-dimensional chemotaxis system with signal-dependent motility and logistic source

Abstract

In this paper, we study the following chemotaxis system with a signal-dependent motility and logistic source: {ut=Δ(γ(v)u)+μu(1−uα),x∈Ω,t>0,0=Δv−v+ur,x∈Ω,t>0,u(x,0)=u0(x),x∈Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} u_{t}=\\Delta {\\bigl(\\gamma (v)u\\bigr)}+\\mu u\\bigl(1-u^{\\alpha}\\bigr), &x \\in \\Omega , t > 0, \\\\ 0=\\Delta v-\\ v+u^{r} , &x\\in \\Omega , t > 0, \\\\ u(x, 0) = u_{0}(x), &x\\in \\Omega \\end{cases} $$\\end{document} under homogeneous Neumann boundary conditions in a smooth bounded domain Omega subset mathbb{R}^{2}, where the motility function gamma (v) satisfies gamma (v)in C^{3}([0,infty )) with gamma (v)>0, and frac{|gamma '(v)|^{2}}{gamma (v)} is bounded for all v > 0. The purpose of this paper is to prove that the model possesses globally bounded solutions. In addition, we show that all solutions (u, v) of the model will exponentially converge to the unique constant steady state (1, 1) as trightarrow +infty when mu geq frac{K}{4^{1+r}} with K=max_{0< vleq infty} frac{|gamma '(v)|^{2}}{gamma (v)}.