The quasilinear Keller–Segel system

endowed with homogeneous Neumann boundary conditions is considered in a bounded domain \(\Omega \subset {\mathbb {R}}^n\), \(n \ge 3\), with smooth boundary for sufficiently regular functions D and S satisfying \(D>0\) on \([0,\infty )\), \(S>0\) on \((0,\infty )\) and \(S(0)=0\). On the one hand, it is shown that if \(\frac{S}{D}\) satisfies the subcritical growth condition

and \(C>0\), then for any sufficiently regular initial data there exists a global weak energy solution such that \({ \mathrm{{ess}}} \sup _{t>0} \Vert u(t) \Vert _{L^p(\Omega )}<\infty \) for some \(p > \frac{2n}{n+2}\). On the other hand, if \(\frac{S}{D}\) satisfies the supercritical growth condition

and \(c>0\), then the nonexistence of a global weak energy solution having the boundedness property stated above is shown for some initial data in the radial setting. This establishes some criticality of the value \(\alpha = \frac{2}{n}\) for \(n \ge 3\), without any additional assumption on the behavior of D(s) as \(s \rightarrow \infty \), in particular without requiring any algebraic lower bound for D. When applied to the Keller–Segel system with volume-filling effect for probability distribution functions of the type \(Q(s) = \exp (-s^\beta )\), \(s \ge 0\), for global solvability the exponent \(\beta = \frac{n-2}{n}\) is seen to be critical.

拟线性凯勒-席格尔系统

赋予齐次诺伊曼边界条件被认为是在有界域\(\Omega \subset {\mathbb {R}}^n\) , \(n \ge 3\)中，对于足够规则的函数D和S满足平滑边界\(D>0\)上\([0,\infty )\)，\(S>0\)上\((0,\infty )\)和\(S(0)=0\)。一方面证明若\(\frac{S}{D}\)满足亚临界生长条件

和\(C>0\)，那么对于任何足够规则的初始数据，都存在一个全局弱能量解，使得\({ \mathrm{{ess}}} \sup _{t>0} \Vert u(t) \Vert _{L^p(\Omega )}<\infty \)对于某些\(p > \frac{2n}{n+2}\)。另一方面，如果\(\frac{S}{D}\)满足超临界生长条件

且\(c>0\)，则对于径向设置中的一些初始数据，显示不存在具有上述有界性质的全局弱能量解。这为\(n \ge 3\)建立了值\(\alpha = \frac{2}{n}\)的一些临界性，而无需对D ( s )的行为作任何额外假设\(s \rightarrow \infty \) ，特别是不需要D的任何代数下界。当应用于具有体积填充效应的 Keller-Segel 系统时，概率分布函数类型为\(Q(s) = \exp (-s^\beta )\)、\(s \ge 0\)，对于全局指数\(\beta = \frac{n-2}{n}\) 的可解性被认为是至关重要的。