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

Abstract

This paper deals with a parabolic–elliptic chemotaxis-consumption system with tensor-valued sensitivity [Formula: see text] under no-flux boundary conditions for [Formula: see text] and Robin-type boundary conditions for [Formula: see text]. The global existence of bounded classical solutions is established in dimension two under general assumptions on tensor-valued sensitivity [Formula: see text]. One of the main steps is to show that [Formula: see text] becomes tiny in [Formula: see text] for every [Formula: see text] and [Formula: see text] when [Formula: see text] is sufficiently small, which seems to be of independent interest. On the other hand, in the case of scalar-valued sensitivity [Formula: see text], there exists a bounded classical solution globally in time for two and higher dimensions provided the domain is a ball with radius [Formula: see text] and all given data are radial. The result of the radial case covers scalar-valued sensitivity [Formula: see text] that can be singular at [Formula: see text].