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

Abstract

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 $r>1$. For a mild logistic dampening, we can improve the smallness condition on the initial datum up to $\sim \frac{1}{1-r}$.

References in corpus (3)

• On a generalized doubly parabolic Keller-Segel system in one spatial dimension
• Exponential Convergence Towards Stationary States for the 1D Porous Medium Equation with Fractional Pressure
• Global existence for some transport equations with nonlocal velocity