In this paper, we study global solutions to the following strongly coupled systems:
{ut=∇⋅(D1∇u−χu∇v)+(a1−b1u−c1v)u,x∈Ω,t>0,0=D2Δv+(a2−b2u−c2v)v,x∈Ω,t>0,\documentclass[12pt]{minimal}
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\begin{document}$$\textstyle\begin{cases} u_{t}=\nabla\cdot(D_{1} \nabla u -\chi u \nabla v)+(a_{1}-b_{1}u-c_{1}v)u,\quad x \in\Omega,t>0, \\ 0=D_{2}\Delta v+(a_{2}-b_{2}u-c_{2}v)v,\quad x \in\Omega,t>0, \end{cases} $$\end{document} over Ω⊂RN\documentclass[12pt]{minimal}
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\begin{document}$\Omega\subset\mathbb{R}^{N}$\end{document}, N≥2\documentclass[12pt]{minimal}
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\begin{document}$N\geq2$\end{document}, subject to homogeneous Neumann boundary conditions and nonnegative initial data. Here Di\documentclass[12pt]{minimal}
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\begin{document}$D_{i}$\end{document}, ai\documentclass[12pt]{minimal}
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\begin{document}$a_{i}$\end{document}, bi\documentclass[12pt]{minimal}
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\begin{document}$b_{i}$\end{document} and ci\documentclass[12pt]{minimal}
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\begin{document}$c_{i}$\end{document}, i=1,2\documentclass[12pt]{minimal}
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\begin{document}$i=1,2$\end{document}, are positive constant. It is proved that this system admits global and bounded classical solutions for all χ>0\documentclass[12pt]{minimal}
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\begin{document}$\chi>0$\end{document}. We also prove the global well-posedness for its repulsive counterpart
{ut=∇⋅(D1∇u+χu∇v)+(a1−b1u−c1v)u,x∈Ω,t>0,0=D2Δv+(a2−b2u−c2v)v,x∈Ω,t>0,\documentclass[12pt]{minimal}
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\begin{document}$$\textstyle\begin{cases} u_{t}=\nabla\cdot(D_{1} \nabla u +\chi u \nabla v)+(a_{1}-b_{1}u-c_{1}v)u,\quad x \in\Omega,t>0, \\ 0=D_{2}\Delta v+(a_{2}-b_{2}u-c_{2}v)v,\quad x \in\Omega,t>0, \end{cases} $$\end{document} provided that b1>a2b2χ(N−2)c2D2N\documentclass[12pt]{minimal}
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\begin{document}$b_{1}>\frac{a_{2}b_{2} \chi(N-2)}{c_{2}D_{2} N}$\end{document}. Our results extend (Discrete Contin. Dyn. Syst. 35:1239–1284, 2015) to higher dimensions and to its repulsive case.
