• DOI:10.12775/TMNA.2016.012
• Corpus ID: 118513684

Boundedness of large-time solutions to a chemotaxis model with nonlocal and semilinear flux

@article{Burczak2014BoundednessOL,
  title={Boundedness of large-time solutions to a chemotaxis model with nonlocal and semilinear flux},
  author={Jan Burczak and Rafael Granero-Belinch'on},
  journal={Topological Methods in Nonlinear Analysis},
  year={2014},
  volume={47},
  pages={369-387}
}

• J. Burczak, Rafael Granero-Belinch'on
• Published 29 September 2014
• Mathematics
• Topological Methods in Nonlinear Analysis

A semilinear version of parabolic-elliptic Keller--Segel system with the \emph{critical} nonlocal diffusion is considered in one space dimension. We show boundedness of weak solutions under very general conditions on our semilinearity. It can degenerate, but has to provide a stronger dissipation for large values of a solution than in the critical linear case or we need to assume certain (explicit) data smallness. Moreover, when one considers a~logistic term with a parameter $r$, we obtain our results even for diffusions slightly weaker than the critical linear one and for arbitrarily large initial datum, provided $r\ge 1$. For a mild logistic dampening, we can improve the smallness condition on the initial datum up to $\sim {1}/({1-r})$.