Renormalized and entropy solutions for the fractional p-Laplacian evolution equations

Abstract

In this paper, we prove the existence and uniqueness of both nonnegative renormalized and entropy solutions for the fractional p-Laplacian evolution problems with nonnegative \(L^1\) data. In addition, we obtain the equivalence of renormalized and entropy solutions and establish a comparison result.

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Acknowledgements

The authors wish to thank the anonymous reviewer for valuable comments and suggestions to improve the expressions. K. Teng was supported by the NSFC (No. 11501403) and the Shanxi Province Science Foundation for Youths (No. 2013021001-3). C. Zhang was supported by the NSFC (No. 11671111) and Heilongjiang Province Postdoctoral Startup Foundation (LBH-Q16082). S. Zhou was supported by the NSFC (Nos. 11571020, 11671021).

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Correspondence to Chao Zhang.

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Teng, K., Zhang, C. & Zhou, S. Renormalized and entropy solutions for the fractional p-Laplacian evolution equations. J. Evol. Equ. 19, 559–584 (2019). https://doi.org/10.1007/s00028-019-00486-9

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Keywords

  • Fractional p-Laplacian
  • Renormalized solutions
  • Entropy solutions
  • Existence
  • Uniqueness

Mathematics Subject Classification

  • Primary 35D05
  • Secondary 35D10
  • 46E35