Continuation of the solution to the chemotaxis problem beyond its blow-up

In this paper we deal with classical solutions to the chemotaxis problem and characterize cases of blow-up. Next, we define continuation of the classical solutions beyond blow-up time T-0. This definition is in fact non-trivial because, although allowing finite mass to concentrate in the origin after T-0 (without prescribing its amount as a function of time), it identifies the solution uniquely. Moreover, two ways of regularizing the problem are provided: they look quite natural and they are shown to converge to the solution.