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APPLIED MATHEMATICS LETTERS, vol.167, 2025 (SCI-Expanded, Scopus)
This article deals with the following singular parabolic-elliptic chemotaxis system { u(t)=Delta u-chi del & sdot;(u/ v del v), x is an element of Omega, (0.1) 0=Delta v-alpha v+mu u, x is an element of Omega, under homogeneous Neumann boundary conditions in a smooth bounded domain Omega subset of R-N with N >= 3, where parameters chi,alpha and mu are positive constants. Fujie, Winkler, and Yokota Fujie(2015) in 2014 and Fujie and Senba Fujie(2016) in 2016 proved that system (0.1) has a unique globally bounded classical solution when alpha=mu=1 and N=2 or chi< 2/N with N >= 3, (0.2) which has remained a critical point for over a decade. However, this article presents a new perspective and shows that assumption (0.2) does not actually constitute a turning point for global classical solutions. Among others, we prove that for all suitable smooth initial data and all alpha,mu>0, the problem (0.1) possesses a global classical solution that is uniformly bounded if chi<2/N+2N-1/ 2N(3)& sdot;root N/ 2N+2 with N >= 3. (0.3) We remark that the current study has enhanced the result (0.2) found in Fujie(2015) and has developed a systematic technique to generate further improvements on the assumption (0.3). Future advancements are intentionally left open for the reader's consideration.