Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system

<jats:p> A family of interpolation inequalities is derived, which differ from estimates of classical Gagliardo–Nirenberg type through the appearance of certain logarithmic deviations from standard Lebesgue norms in zero-order expressions. Optimality of the obtained inequalities is shown. A subsequent application reveals that when posed under homogeneous Neumann boundary conditions in smoothly bounded planar domains and with suitably regular initial data, for any choice of <jats:inline-formula> <jats:tex-math>\alpha&gt;0</jats:tex-math> </jats:inline-formula> the Keller–Segel-type migration–consumption system <jats:inline-formula> <jats:tex-math>u_{t} = \Delta (uv^{-\alpha})</jats:tex-math> </jats:inline-formula> , <jats:inline-formula> <jats:tex-math>v_{t} = \Delta v-uv</jats:tex-math> </jats:inline-formula> , admits a global classical solution. </jats:p>