In a smoothly bounded domain \(\Omega \subset \mathbb {R}^n\), \(n\ge 1\), the quasilinear Keller–Segel system

is considered under homogeneous no-flux boundary conditions. It is firstly shown that if D and S, besides belonging to \(C^2([0,\infty ))\) with \(S(0)=0\), merely satisfy

then for all \(K>0\) there exists \(\varepsilon _\star (K)\in (0,\frac{R}{2})\) such that whenever \(0\le u_0\in W^{1,\infty }(\Omega )\) and \(0\le v_0\in W^{1,\infty }(\Omega )\) satisfy

a corresponding initial value problem for (\(\star \)) admits a global bounded classical solution with \((u,v)|_{t=0}=(u_0,v_0)\). Secondly, a more restrictive condition on the initial data, inter alia requiring appropriate smallness of both \(\Vert u_0\Vert _{L^\infty (\Omega )}\) and \(\Vert v_0\Vert _{W^{1,\infty }(\Omega )}\), is identified as sufficient to ensure exponential stabilization of the correspondingly obtained solution toward the equilibrium \((\frac{1}{|\Omega |} \int _\Omega u_0, \frac{1}{|\Omega |}\int _\Omega u_0)\). As a technical ingredient of crucial importance for the derivation of explicit pointwise bounds for the respective first solution components, the analysis relies on a refinement of a Moser-type iterative argument which, formulated here in a general context of parabolic inequalities, provides some quantitative information about the dependence of \(L^\infty \) estimates on bounds on the initial data and \(L^1\) bounds.

在光滑有界域\(\Omega \subset \mathbb {R}^n\) , \(n\ge 1\) 中，拟线性 Keller-Segel 系统

是在均匀无通量边界条件下考虑的。首先证明如果D和S，除了属于\(C^2([0,\infty ))\)和\(S(0)=0\)，仅仅满足

那么对于所有\(K>0\)存在\(\varepsilon _\star (K)\in (0,\frac{R}{2})\)使得每当\(0\le u_0\in W ^{1,\infty }(\Omega )\)和\(0\le v_0\in W^{1,\infty }(\Omega )\)满足

( \(\star \) )的相应初始值问题承认具有\((u,v)|_{t=0}=(u_0,v_0)\)的全局有界经典解。其次，对初始数据的限制性更强，特别是需要\(\Vert u_0\Vert _{L^\infty (\Omega )}\)和\(\Vert v_0\Vert _{W^ {1,\infty }(\Omega )}\)，被确定为足以确保相应获得的解决方案的指数稳定朝着平衡\((\frac{1}{|\Omega |} \int _\Omega u_0 , \frac{1}{|\Omega |}\int _\Omega u_0)\). 作为推导各个第一解分量的显式逐点边界至关重要的技术要素，该分析依赖于对 Moser 型迭代论证的改进，该论证在此处在抛物线不等式的一般背景下制定，提供了一些定量信息关于\(L^\infty \)估计对初始数据边界和\(L^1\)边界的依赖性。