On uniqueness theorem on weak solutions to the parabolic-parabolic Keller-Segel system of degenerate and singular types

Masanari Miura, Yoshie Sugiyama

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

The uniqueness of weak solutions to the parabolic-parabolic Keller-Segel systems (KS)m below with m>max {1/2-1/n,0} is proved in the class of Hölder continuous functions for any space dimension n. Since Hölder continuity is an optimal regularity for weak solutions of the porous medium equation, it seems to be reasonable to investigate its uniqueness in such a class of solutions. Our proof is based on the standard duality argument coupled with vanishing viscosity method which recovers degeneracy for m>1, and which removes singularities for 0<m<1 in the energy class of solutions.

Original languageEnglish
Pages (from-to)4064-4086
Number of pages23
JournalJournal of Differential Equations
Volume257
Issue number11
DOIs
Publication statusPublished - Dec 1 2014

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Uniqueness Theorem
Weak Solution
Porous materials
Viscosity
Uniqueness
Viscosity Method
Vanishing Viscosity
Porous Medium Equation
Degeneracy
Continuous Function
Duality
Regularity
Singularity
Energy
Class

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

On uniqueness theorem on weak solutions to the parabolic-parabolic Keller-Segel system of degenerate and singular types. / Miura, Masanari; Sugiyama, Yoshie.

In: Journal of Differential Equations, Vol. 257, No. 11, 01.12.2014, p. 4064-4086.

Research output: Contribution to journalArticle

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