Parabolic elliptic type Keller-Segel system on the whole space case

Jinhuan  Wang; Li Chen; Liang  Hong

doi:10.3934/dcds.2016.36.1061

Article Contents

2016, Volume 36, Issue 2: 1061-1084. Doi: 10.3934/dcds.2016.36.1061

This issue Previous Article Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy Next Article On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum

Parabolic elliptic type Keller-Segel system on the whole space case

1.
School of Mathematics, Liaoning University, Shenyang 110036
2.
Universität Mannheim, Lehrstuhl für Mathematik IV, 68131, Mannheim

Received: July 2014

Published: May 2017

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Abstract

This note is devoted to the discussion on the existence and blow up of the solutions to the parabolic elliptic type Patlak-Keller-Segel system on the whole space case. The problem in two dimension is closely related to the Logarithmic Hardy-Littlewood-Sobolev inequality, which directly introduced the critical mass $8\pi$. While in the higher dimension case, it is related to the Hardy-Littlewood-Sobolev inequality. Therefore, a porous media type nonlinear diffusion has been introduced in order to balance the aggregation. We will review the critical exponents which were introduced in the literature, namely, the exponent $m=2-2/n$ which comes from the scaling invariance of the mass, and the exponent $m=2n/(n+2)$ which comes from the conformal invariance of the entropy. Finally a new result on the model with a general potential, inspired from the Hardy-Littlewood-Sobolev inequality, will be given.

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Mathematics Subject Classification: Primary: 35K65, 35B45, 35J20.

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