Global well-posedness in a three-dimensional chemotaxis-consumption model with singular sensitivity

In this paper, we present a three-dimensional version of a dissipative chemotactic system with singular sensitivity coefficients, nt=Delta n-chi del<middle dot>(n del lnc),ct=Delta c-nc.$$\begin{align*} \hspace*{105pt} \left\lbrace \def\eqcellsep{&}\begin{array}{ll} n_{t}=\Delta n-\chi \nabla \cdot (n\nabla {\ln c}), \\ c_{t}=\Delta c-nc. \end{array} \right. \end{align*}$$Thus far, the comprehensive characterization of the impact of singular sensitivity on the existence of classical solutions in three-dimensional systems remains incomplete due to the singularity effect. Even in two-dimensional space, only global generalized solutions have been obtained for general initial values while results related to small initial values, are available for ensuring the global existence of classical solutions. In this direction, under some smallness assumptions on initial values, the present work provides some interesting contributions by establishing results on global boundedness by using the energy functional method, and especially on large time stabilization toward homogeneous equilibria.