## Dynamic and Steady States for Multi-Dimensional Keller-Segel Model with Diffusion Exponent m > 0

This paper investigates infinite-time spreading and finite-time blow-up for the Keller-Segel system. For 0 < m ≤ 2 - 2/d, the L p space for both dynamic and steady solutions are detected with (Formula presented.). Firstly, the global existence of the weak solution is proved for small initial data in L p. Moreover, when m > 1 - 2/d, the weak solution preserves mass and satisfies the hyper-contractive estimates in L q for any p < q < ∞. Furthermore, for slow diffusion 1 < m ≤ 2 - 2/d, this weak solution is also a weak entropy solution which blows up at finite time provided by the initial negative free energy. For m > 2 - 2/d, the hyper-contractive estimates are also obtained. Finally, we focus on the L p norm of the steady solutions, it is shown that the energy critical exponent m = 2d/(d + 2) is the critical exponent separating finite L p norm and infinite L p norm for the steady state solutions. © 2013 Springer-Verlag Berlin Heidelberg.

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## Related Subject Headings

- Mathematical Physics
- 5107 Particle and high energy physics
- 4904 Pure mathematics
- 4902 Mathematical physics
- 0206 Quantum Physics
- 0105 Mathematical Physics
- 0101 Pure Mathematics

### Citation

*Communications in Mathematical Physics*,

*323*(3), 1017–1070. https://doi.org/10.1007/s00220-013-1777-z

*Communications in Mathematical Physics*323, no. 3 (November 1, 2013): 1017–70. https://doi.org/10.1007/s00220-013-1777-z.

*Communications in Mathematical Physics*, vol. 323, no. 3, Nov. 2013, pp. 1017–70.

*Scopus*, doi:10.1007/s00220-013-1777-z.

## Published In

## DOI

## EISSN

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- Mathematical Physics
- 5107 Particle and high energy physics
- 4904 Pure mathematics
- 4902 Mathematical physics
- 0206 Quantum Physics
- 0105 Mathematical Physics
- 0101 Pure Mathematics