Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions

By Adrien Blanchet, José Antonio Carrillo, Philippe Laurençot. 2008

219 citations recorded by the source corpus.

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak–Keller–Segel system with d ≥ 3 and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass Mc such that if $${M \in (0, M_c]}$$ solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for $${M \in (0, M_c)}$$ . While characterising the possible infinite time blowing-up profile for M = Mc, we observe that the long time asymptotics are much more complicated than in the classical Patlak–Keller–Segel system in dimension two.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/61553fbe8c2149d632852c1a50a5d73524490492

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Citations in the corpus

Equilibria of homogeneous functionals in the fair-competition regime (2016, methodology): “The role of both non-linear diffusion and non-local aggregation terms was clarified in [7], see also [50], where the authors find that there is a similar dichotomy to the two-dimensional classical Keller–Segel case (N = 2,m = 1, k = 0), for a whole range of parameters, choosing the non-local term as”
Refined asymptotics for the subcritical Keller-Segel system and related functional inequalities (2010, unknown): “A P] 1”
Uniqueness for Keller-Segel-type chemotaxis models (2012, result): “We finally refer to [11, 13, 14, 22] for different results concerning the existence of solutions satisfying the boundedness of the Fisher information and/or the uniform bounds of the solutions for particular choices of ε ≥ 0, α ≥ 0, and nonlinear diffusions.”
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system (2011, result): “As opposed to this, the few results addressing blow-up also in the setting of nonlinear diffusion concentrate on parabolic-elliptic simplifications (see [5], [6], [3], [8], [10], for instance) or on unboundedness in possibly infinite time ([19], [35]), the apparently only exception being a recent re”
Existence, Uniqueness and Lipschitz Dependence for Patlak–Keller–Segel and Navier–Stokes in $${\mathbb{R}^2}$$R2 with Measure-Valued Initial Data (2012, background): “1) involving nonlinear diffusion which are critical in higher dimensions have also been studied [51, 14, 65, 66, 10, 50] (see also the related [53]).”
Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data (2012, methodology): “1) involving nonlinear diffusion which are critical in higher dimensions have also been studied [10,14,50,51,65,66] (see also the related [53]).”
How strong a logistic damping can prevent blow-up for the minimal Keller–Segel chemotaxis system? (2016, background): “4, especially, (P5) and (P8) suggest certain clues on how to produce unbounded solutions [4, 9, 10, 28, 54, 55].”
On minimizers of interaction functionals with competing attractive and repulsive potentials (2015, unknown): “[7, 8, 14, 17, 20, 21, 22, 23]).”