The Keller-Segel system has been widely validated as a model for cell movement under the combined effect of stochastic diffusion and a directed drift due to a chemoattractant emitted by the cells themselves (chemotaxis). In two dimensions, and in its simpler mathematical setting, it has been proved that the system admits a critical mass (below this mass there are global solutions, above the system blows-up). In higher dimensions this is not true and L d/2 is the critical norm but only less precise statements are available. Here, we study a variant for the diffusion law of the chemical potential in chemotaxis, based on a convolution operator or in other words on a fractional diffusion law. The aim of this paper is to enumerate the corresponding qualitative properties-global existence, blow up, stationary states-for the model with critical logarithmic kernel. In particular we show that this new system leads to a unified theory with critical mass in any dimension.
Abstract. In this paper, we study a variation of the equations of a chemotaxis kinetic model and investigate it in one dimension. In fact, we use fractional diffusion for the chemoattractant in the Othmar-Dunbar-Alt system (Othmer in J Math Biol 26(3): 1988). This version was exhibited in Calvez in Amer Math Soc, pp [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62] 2007 for the macroscopic well-known Keller-Segel model in all space dimensions. These two macroscopic and kinetic models are related as mentioned in Bournaveas, Ann Inst H Poincaré
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