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arXiv · 1409.8102

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

Jan Burczak·Rafael Granero-Belinchón
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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 $\sim \frac{1}{1-r}$.

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Jan Burczak, Rafael Granero-Belinchón. 2014-09-29. Boundedness of large-time solutions to a chemotaxis model with nonlocal and semilinear flux. https://arxiv.org/abs/1409.8102

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