Regular solutions of chemotaxis-consumption systems involving tensor-valued sensitivities and Robin type boundary conditions

TL;DR

This paper proves the global existence of bounded classical solutions for a chemotaxis-consumption system with tensor-valued sensitivities under no-flux and Robin boundary conditions, highlighting key regularity properties and special radial cases.

Contribution

It establishes the global existence of solutions for tensor-valued sensitivities and extends results to scalar sensitivities with singularities in radial domains.

Findings

Global bounded solutions exist in 2D for tensor-valued sensitivities.

Radial solutions exist in higher dimensions even with singular sensitivities.

Gradient of chemoattractant becomes small in localized regions over time.

Abstract

This paper deals with a parabolic-elliptic chemotaxis-consumption system with tensor-valued sensitivity $S(x,n,c)$ under no-flux boundary conditions for $n$ and Robin-type boundary conditions for $c$. The global existence of bounded classical solutions is established in dimension two under general assumptions on tensor-valued sensitivity $S$. One of main steps is to show that $\nabla c(\cdot,t)$ becomes tiny in $L^{2}(B_{r}(x)\cap \Omega)$ for every $x\in\overline{\Omega}$ and $t$ when $r$ is sufficiently small, which seems to be of independent interest. On the other hand, in the case of scalar-valued sensitivity $S=\chi(x,n,c)\mathbb{I}$, there exists a bounded classical solution globally in time for two and higher dimensions provided the domain is a ball with radius $R$ and all given data are radial. The result of the radial case covers scalar-valued sensitivity $\chi$ that can be singular at $c=0$.