Preventing $L^p$ blow-up by local anisotropy of signal production in the Keller-Segel system with strongly differing diffusion rates

Abstract

In a smoothly bounded domain $\Omega \subset R^n$, $n\le 5$, the manuscript considers the variant of the Keller-Segel system given by \[ \left\{ \begin{array}{l} u_t = D Δu - \nabla \cdot (u\nabla v), \\[1mm] v_t = d Δv + \nabla \cdot (u\nabla v) - v + u, \end{array} \right. \] which involves an additional contribution $\nabla \cdot (u\nabla v)$ to the chemoattractant evolution, in line with refined modeling literature reflecting an anisotropic correction to the isotropic signal production term $+u$ in the classical Keller-Segel model. It is shown that for arbitrary $D>0$ and $d>0$ and any nonnegative intial data from $W^{1,\infty }(\Omega )\times W^{1,\infty }(\Omega )$, an associated Neumann problem admits a global weak solution $(u,v)$ which, inter alia, satisfies \[ \sup_{t \in (0,\infty)\setminus N} \int_Ωe^{u^α(\cdot,t)} < \infty \] with some $\alpha >0$ and some null set $N\subset (0,\infty )$.