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The $L^1$-critical parabolic-elliptic Patlak--Keller--Segel system is a classical model of chemotactic aggregation in microorganisms well known to have critical mass phenomena [Blanchet, Dolbeault, and Perthame, Electron. J. Differential Equations, 2006 (2006), pp. 1--23; Blanchet, Carrillo, and Laurencot, Calc. Var., 35 (2009), pp. 133--168]. In this paper we study this critical mass phenomenon in the context of Patlak--Keller--Segel models with spatially varying diffusivity of the chemoattractant. The primary issue is how, if possible, one localizes the presence of the inhomogeneity in the nonlocal term. Our methods also provide new blow-up results for the homogeneous problem with nonlinear diffusion, showing that there exist blow-up solutions with arbitrarily large (positive) initial free energy.

Global Existence and Finite Time Blow-Up for Critical Patlak--Keller--Segel Models with Inhomogeneous Diffusion