A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: Global solvability for large nonradial data

The chemotaxis system $$\{\begin{array}{l}{u}_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (\frac{u}{v}\mathrm{\nabla }v),v_t=\mathrm{\Delta }v-v+u,\end{array}$$ is considered in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with smooth boundary, where $\chi >0$. An apparently novel type of generalized solution framework is introduced within which an extension of previously known ranges for the key parameter $\chi$ with regard to global solvability is achieved. In particular, it is shown that under the hypothesis that$$\chi <\{\begin{array}{llll}\mathrm{\infty }& ifn=2,\sqrt{8}& ifn=3,\frac{n}{n-2}& ifn\ge 4,\end{array}$$ for all initial data satisfying suitable assumptions on regularity and positivity, an associated no-flux initial-boundary value problem admits a globally defined generalized solution. This solution inter alia has the property that $$u\in L_{loc}^1(\bar{\mathrm{\Omega }}\times [0,\mathrm{\infty }))\mathrm{.}$$