On a quasilinear fully parabolic two-species chemotaxis system with two chemicals

Abstract

This paper deals with the following two-species chemotaxis system with nonlinear diffusion, sensitivity, signal secretion and (without or with) logistic source \\begin{document}$ \\begin{eqnarray*} \\left\\{ \\begin{array}{llll} u_t = \\nabla \\cdot (D_1(u)\\nabla u - S_1(u)\\nabla v) + f_{1}(u),\\quad x >under homogeneous Neumann boundary conditions in a bounded domain \\begin{document}$ \\Omega\\subset \\mathbb{R}^n $\\end{document} with \\begin{document}$ n\\geq2 $\\end{document} . The diffusion functions \\begin{document}$ D_{i}(s) \\in C^{2}([0,\\infty)) $\\end{document} and the chemotactic sensitivity functions \\begin{document}$ S_{i}(s) \\in C^{2}([0,\\infty)) $\\end{document} are given by \\begin{document}$ \\begin{equation*} \\begin{split} D_{i}(s) \\geq C_{d_{i}} (1+s)^{-\\alpha_i} \\quad \\text{and} \\quad 0 where \\begin{document}$ C_{d_{i}},C_{s_{i}}>0 $\\end{document} and \\begin{document}$ \\alpha_i,\\beta_{i} \\in \\mathbb{R} $\\end{document} \\begin{document}$ (i = 1,2) $\\end{document} . The logistic source functions \\begin{document}$ f_{i}(s) \\in C^{0}([0,\\infty)) $\\end{document} and the nonlinear signal secretion functions \\begin{document}$ g_{i}(s) \\in C^{1}([0,\\infty)) $\\end{document} are given by \\begin{document}$ \\begin{equation*} \\begin{split} f_{i}(s) \\leq r_{i}s - \\mu_{i} s^{k_{i}} \\quad \\text{and} \\quad g_{i}(s)\\leq s^{\\gamma_{i}} \\text{ for all } s\\geq0, \\end{split} \\end{equation*} $\\end{document} where \\begin{document}$ r_{i} \\in \\mathbb{R} $\\end{document} , \\begin{document}$ \\mu_{i},\\gamma_{i} > 0 $\\end{document} and \\begin{document}$ k_{i} > 1 $\\end{document} \\begin{document}$ (i = 1,2) $\\end{document} . With the assumption of proper initial data regularity, the global boundedness of solution is established under the some specific conditions with or without the logistic functions \\begin{document}$ f_{i}(s) $\\end{document} . Moreover, in case \\begin{document}$ r_{i}>0 $\\end{document} , for the large time behavior of the smooth bounded solution, by constructing the appropriate energy functions, under the conditions \\begin{document}$ \\mu_{i} $\\end{document} are sufficiently large, it is shown that the global bounded solution exponentially converges to \\begin{document}$ \\left((\\frac{r_{1}}{\\mu_{1}})^{\\frac{1}{k_{1}-1}}, (\\frac{r_{2}}{\\mu_{2}})^{\\frac{\\gamma_{1}}{k_{2}-1}}, (\\frac{r_{2}}{\\mu_{2}})^{\\frac{1}{k_{2}-1}}, (\\frac{r_{1}}{\\mu_{1}})^{\\frac{\\gamma_{2}}{k_{1}-1}}\\right) $\\end{document} as \\begin{document}$ t\\rightarrow\\infty $\\end{document} .