Suppression of blow-up by local anisotropy of signal production in the Keller-Segel system

Abstract

In a smoothly bounded domain $\Omega \subset R^n$, $n\le 5$, and with $D>0$ and $d>0$, this manuscript considers the Neumann initial-boundary problem for the Keller-Segel type system$\{\begin{array}{c}{u}_t=D\Delta u-\nabla \cdot \left(u\nabla v\right),\\ v_t=d\Delta v+\nabla \cdot \left(u\nabla v\right)-v+u,\end{array}(\star )$ which arises in the modeling for chemotactic movement in the presence of certain anisotropic signal production mechanisms.

Unlike the classical Keller-Segel model whose solutions may blow up in finite time in high-dimensional domains, this problem is shown to admit a unique global bounded classical solution whenever the difference $|D-d|$ is appropriately small. This markedly distinguishes (⋆) from classical Keller-Segel systems for which some solutions are known to blow up in finite time when $n\ge 2$.