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

초록

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(Br(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. © World Scientific Publishing Company.

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