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manuscripta mathematica

, Volume 159, Issue 3–4, pp 475–509 | Cite as

Finite time blow up and non-uniform bound for solutions to a degenerate drift-diffusion equation with the mass critical exponent under non-weight condition

  • Takayoshi Ogawa
  • Hiroshi WakuiEmail author
Article
  • 62 Downloads

Abstract

We consider the non-existence and the non-uniform boundedness of a time global solution to the Cauchy problem of a degenerate drift-diffusion system with the mass critical exponent. If the initial data has negative free energy, then either the corresponding weak solution to the equation does not exist globally in time, or the time global solution does not remain bounded in the energy space. We emphasize that our result does not require any weight assumption on the initial data, and hence, a solution may have an infinite second moment. The proof is based upon the modified virial law and conservation laws and we show that the modified moment functional vanishes for a finite time under the negative energy condition. For a radially symmetric case, the solution blows up in finite time and the mass concentration phenomenon occurs with a sharp lower bound related to the best constant for the Hardy–Littlewood–Sobolev inequality.

Mathematics Subject Classification

Primary 35K65 Secondary 35K45 35B33 35B44 

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Notes

Acknowledgements

The first author is partially supported by JSPS Grant-in-aid for Scientific Research S #25220702 and Challenging Research (Pioneering) #17H06199. The Second author is supported by JSPS Grant-in-aid for Scientific Research S #25220702.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematical InstituteTohoku UniversitySendaiJapan

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