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 tensorvalued 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 the main steps is to show that del c(circle; t) becomes tiny in L-2(Br(x)boolean AND Omega) for every x is an element of (sic) 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)(sic)ere 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.