Relaxation in a Keller-Segel-consumption system involving signal-dependent motilities
Genglin Li, Michael Winkler
Abstract
Two relaxation features of the migration-consumption chemotaxis system involving signal-dependent motilities, ), then for any initial data (u 0 ,v 0 ) belonging to the space (C 0 (Ω)) ⋆ × L ∞ (Ω) an associated no-flux type initial-boundary value problem admits a global very weak solution.Beyond this initial relaxation property, it is seen that under the additional hypotheses that ϕ ∈ C 1 ([0,∞)) and n ≤ 3, each of these solutions stabilizes toward a semi-trivial spatially homogeneous steady state in the large time limit.By thus applying to irregular and partially even measure-type initial data of arbitrary size, this firstly extends previous results on global solvability in (⋆) which have been restricted to initial data not only considerably more regular but also suitably small.Secondly, this reveals a significant difference between the large time behavior in (⋆) and that in related degenerate counterparts involving functions ϕ with ϕ(0) = 0, about which, namely, it is known that some solutions may asymptotically approach nonhomogeneous states.