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

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 the main steps is to show that ∇c(·, t) becomes tiny in L²(B_r(x) ∩ ω¯) for every x ω¯ 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 = χ(x,n,c)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 χ that can be singular at c = 0.