This paper deals with the global existence for a class of Keller–Segel model with signal-dependent motility and general logistic term under homogeneous Neumann boundary conditions in a higher-dimensional smoothly bounded domain, which can be written as $$\eqalign{& u_t = \Delta (\gamma (v)u) + \rho u-\mu u^l,\quad x\in \Omega ,\;t > 0, \cr & v_t = \Delta v-v + u,\quad x\in \Omega ,\;t > 0.} $$It is shown that whenever ρ ∈ ℝ, μ > 0 and $$l > \max \left\{ {\displaystyle{{n + 2} \over 2},2} \right\},$$then the considered system possesses a global classical solution for all sufficiently smooth initial data. Furthermore, the solution converges to the equilibrium $$\left( {{\left( {\displaystyle{{\rho _ + } \over \mu }} \right)}^{1/(l-1)},{\left( {\displaystyle{{\rho _ + } \over \mu }} \right)}^{1/(l-1)}} \right)$$as t → ∞ under some extra hypotheses, where ρ+ = max{ρ, 0}.

中文翻译：

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具有信号依赖运动和广义逻辑源的n维趋化系统：全局存在和渐近稳定

本文研究了一类具有信号相关运动和一般逻辑项的 Keller-Segel 模型在高维平滑有界域中齐次 Neumann 边界条件下的全局存在性，可写为$$\eqalign{& u_t = \Delta (\gamma (v)u) + \rho u-\mu u^l,\quad x\in \Omega ,\;t > 0, \cr & v_t = \Delta vv + u,\quad x\in \Omega ,\;t > 0.} $$表明每当ρ∈ ℝ,μ> 0 和$$l > \max \left\{ {\displaystyle{{n + 2} \over 2},2} \right\},$$那么所考虑的系统对所有足够平滑的初始数据都具有全局经典解。此外，解收敛于平衡$$\left( {{\left( {\displaystyle{{\rho _ + } \over \mu }} \right)}^{1/(l-1)},{\left( {\displaystyle{{ \rho _ + } \over \mu }} \right)}^{1/(l-1)}} \right)$$作为吨→ ∞ 在一些额外假设下，其中ρ+=最大{ρ, 0}。