The 8π‐problem for radially symmetric solutions of a chemotaxis model in the plane

By Piotr Biler, Grzegorz Karol Karch, Philippe Laurençot, T. Nadzieja. 2006

173 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/81174723e21e47e0e18ca1a2236a6cf5dc10ee6f

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In Applied Mathematics

In Mathematical Physics

In Modeling and Simulation

Citations in the corpus

Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions (2008, background): “This is in sharp contrast with the two-dimensional PKS case where infinite time blow-up always occurs, see [2, 3].”
Global existence for the Cauchy problem of the parabolic–parabolic Keller–Segel system on the plane (2012, unknown): “If m(u0; R2) = 8π , then the solution of (2) exists globally in time [4,5].”
Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane (2014, methodology): “Indeed, following [7], let y0 := sup{y > 0 such that φ1(z) > φ2(z) for all 0 < z < y} .”
Uniqueness and Long Time Asymptotic for the Keller–Segel Equation: The Parabolic–Elliptic Case (2013, background): “22) has been proved in [8, 5], see also [15, 16, 17].”
The parabolic-parabolic Keller-Segel model in R2 (2008, background): “when the initial mass M equals the threshold value ([5, 3]).”
Functional inequalities, thick tails and asymptotics for the critical mass Patlak–Keller–Segel model (2010, unknown): “Mathematics subject classification numbers: 15A45, 49M20”
Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion (2010, background): “See [9, 14, 27] for the corresponding result for classical 2D PKS.”
Blow-up, Concentration Phenomenon and Global Existence for the Keller–Segel Model in High Dimension (2010, unknown): “t→∞, [8, 12].”
Existence, Uniqueness and Lipschitz Dependence for Patlak–Keller–Segel and Navier–Stokes in $${\mathbb{R}^2}$$R2 with Measure-Valued Initial Data (2012, background): “show in [7] that for all α ∈ (0, 8π) there exists a unique, radially symmetric self-similar solution with mass α, denoted here in self-similar variables by Gα(ξ) (existence had been previously established in [5, 56]).”
On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher (2011, unknown): “Using the comparison principle in the radial variable, it has been proven in [7] that the radial, non-negative smooth solution to this problem is unique.”
Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data (2012, background): “1 in [7] to prove that n′(0) exists and is finite, and hence Eα and e are bounded at zero.”