On uniqueness theorem on weak solutions to the parabolic-parabolic Keller-Segel system of degenerate and singular types

The uniqueness of weak solutions to the parabolic-parabolic Keller-Segel systems (KS)_m below with m>max {1/2-1/n,0} is proved in the class of Hölder continuous functions for any space dimension n. Since Hölder continuity is an optimal regularity for weak solutions of the porous medium equation, it seems to be reasonable to investigate its uniqueness in such a class of solutions. Our proof is based on the standard duality argument coupled with vanishing viscosity method which recovers degeneracy for m>1, and which removes singularities for 0<m<1 in the energy class of solutions.