Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension

This manuscript considers the no-flux initial-boundary problem for the migration-consumption system {u(t) = Delta(u phi(v)), (star) v(t) = Delta v - uv, in a smoothly bounded domain Omega subset of R-n, n >= 1, where phi suitably generalizes the singular prototype given by phi(xi) = xi(-alpha), xi > 0, with alpha > 0. It is firstly shown that for such diffusion singularities of arbitrary strength, and for any given initial data u(0) and v(0) from W-1,W-infinity (Omega) with u(0) >= 0 and v(0) > 0 in (Omega) over bar, a so-called very weak-strong solution (u, v) with (u, v)vertical bar(t=0)=(u(0), v(0)) can be constructed. Under the additional restrictions that 2 <= n <= 5 and alpha > n-2/6-n, and under an additional assumption on f phi', it is furthermore asserted that the flux del(u phi(v)) belongs to some reflexive Lebesgue space, and that (star) is satisfied in a standard weak sense. Finally, in the one-dimensional case a statement on global classical solvability is derived under the mere condition that phi is an element of C-3((0, infinity)) be positive. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This manuscript considers the no-flux initial-boundary problem for the migration-consumption system and provides the existence of a weak-strong solution for arbitrary diffusion singularities with suitable initial data. It also presents the conditions for the flux del(u phi(v)) to belong to a reflexive Lebesgue space and for the system to be satisfied in a weak sense. Finally, it derives a statement on global classical solvability in the one-dimensional case under certain conditions on phi.