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

Publication ,  Journal Article
Bian, S; Liu, JG
Published in: Communications in Mathematical Physics
November 1, 2013

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.

Duke Scholars

Published In

Communications in Mathematical Physics

DOI

EISSN

1432-0916

ISSN

0010-3616

Publication Date

November 1, 2013

Volume

323

Issue

3

Start / End Page

1017 / 1070

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

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Bian, S., & Liu, J. G. (2013). Dynamic and Steady States for Multi-Dimensional Keller-Segel Model with Diffusion Exponent m > 0. Communications in Mathematical Physics, 323(3), 1017–1070. https://doi.org/10.1007/s00220-013-1777-z
Bian, S., and J. G. Liu. “Dynamic and Steady States for Multi-Dimensional Keller-Segel Model with Diffusion Exponent m > 0.” Communications in Mathematical Physics 323, no. 3 (November 1, 2013): 1017–70. https://doi.org/10.1007/s00220-013-1777-z.
Bian S, Liu JG. Dynamic and Steady States for Multi-Dimensional Keller-Segel Model with Diffusion Exponent m > 0. Communications in Mathematical Physics. 2013 Nov 1;323(3):1017–70.
Bian, S., and J. G. Liu. “Dynamic and Steady States for Multi-Dimensional Keller-Segel Model with Diffusion Exponent m > 0.” Communications in Mathematical Physics, vol. 323, no. 3, Nov. 2013, pp. 1017–70. Scopus, doi:10.1007/s00220-013-1777-z.
Bian S, Liu JG. Dynamic and Steady States for Multi-Dimensional Keller-Segel Model with Diffusion Exponent m > 0. Communications in Mathematical Physics. 2013 Nov 1;323(3):1017–1070.
Journal cover image

Published In

Communications in Mathematical Physics

DOI

EISSN

1432-0916

ISSN

0010-3616

Publication Date

November 1, 2013

Volume

323

Issue

3

Start / End Page

1017 / 1070

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