Hölder regularity and uniqueness theorem on weak solutions to the degenerate Keller-Segel system (Q272750)

The Keller-Segel model for chemotaxis NEWLINE\[NEWLINEu_t=\nabla\cdot(\nabla u^m-u^{q-1}\nabla v),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\delta v_t=\Delta v-\gamma v+u,NEWLINE\]NEWLINE is studied in the whole space \(\mathbb R^n\) when \(m>1\) and \(q>2\). The main goal is to prove uniform Hölder estimates for weak solutions depending on the supremum of the density function and modified energy estimates, and intrinsic scales considered in the porous medium equation. As a consequence of those Hölder regularity properties and the \(L^1\)-contraction property, the uniqueness of weak solutions is obtained for \(q>\max(1+m/2,2)\).