GLOBAL EXISTENCE AND LARGE TIME BEHAVIOR OF A 2D KELLER-SEGEL SYSTEM IN LOGARITHMIC LEBESGUE SPACES

This paper is devoted to the global analysis for the two-dimensional parabolic-parabolic Keller-Segel system in the whole space. By well balanced arguments of the L-1 and L-infinity spaces, we first prove global well-posedness of the system in L-1 x L-infinity which partially answers the question posted by Kozono et al in [19]. For the case mu(0) > 0, we make full use of the linear parts of the system to get the improved long time decay property. Moreover, by using the new formulation involving all linear parts, introducing the logarithmic-weight in time to modify the other endpoint space L-infinity x L-infinity, and carefully decomposing time into several pieces, we are able to establish the global well-posedness and large time behavior of the system in L-ln(infinity) x L-infinity.