Supercritical degenerate parabolic–parabolic Keller–Segel system: existence criterion given by the best constant in Sobolev’s inequality

  • Jinhuan Wang
  • Yue Li
  • Li ChenEmail author


This article presents a relationship between the sharp constant of the Sobolev inequality and the initial criterion to the global existence of degenerate parabolic–parabolic Keller–Segel system with the diffusion exponent \(\frac{2n}{2+n}<m<2-\frac{2}{n}\). The global weak solution obtained in this article does not need any smallness assumption on the initial data. Furthermore, a uniform in time \(L^{\infty }\) estimate of the weak solutions is obtained via the Moser iteration, where the constant in \(L^p\) estimate for the gradient of the chemical concentration has been exactly formulated in order to complete the iteration process.


Degenerate Keller–Segel system Supercritical exponent \(L^\infty \) estimate Free energy 

Mathematics Subject Classification

Primary 35B44 Secondary 35K55 



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Authors and Affiliations

  1. 1.School of MathematicsLiaoning UniversityShenyangPeople’s Republic of China
  2. 2.Department of MathematicsNanjing UniversityNanjingPeople’s Republic of China
  3. 3.Lehrstuhl für Mathematik IVUniversität MannheimMannheimGermany

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