No critical nonlinear diffusion in 1d quasilinear fully parabolic chemotaxis system

Tomasz Cieślak, Kentarou Fujie

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


This paper deals with the fully parabolic 1d chemotaxis system with diffusion 1/(1 + u). We prove that the above mentioned nonlinearity, despite being a natural candidate, is not critical. It means that for such a diffusion any initial condition, independently on the magnitude of mass, generates the global-in-time solution. In view of our theorem one sees that the one-dimensional Keller-Segel system is essentially different from its higher-dimensional versions. In order to prove our theorem we establish a new Lyapunov-like functional associated to the system. The information we gain from our new functional (together with some estimates based on the well-known classical Lyapunov functional) turns out to be rich enough to establish global existence for the initial-boundary value problem.

Original languageEnglish
Pages (from-to)2529-2540
Number of pages12
JournalProceedings of the American Mathematical Society
Issue number6
Publication statusPublished - 2018


  • Chemotaxis
  • Global existence
  • Lyapunov functional


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