Continuation of the Solution to the Chemotaxis Problem Beyond its Blow-up

Abstract

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_o. This definition is in fact nontrivial because, although allowing finite mass to concentrate in the origin after T_o (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.