Relaxation in a Keller-Segel-consumption system involving signal-dependent motilities

By Genglin Li, Michael Winkler. Communications in Mathematical Sciences, 2022

47 citations recorded by the source corpus.

Two relaxation features of the migration-consumption chemotaxis system involving signal-dependent motilities, are studied in smoothly bounded domains Ω ⊂ R n , n ≥ 1: It is shown that if φ ∈ C 0 ([0 , ∞ )) is positive on [0 , ∞ ), 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.

Subfields: Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/cb1c34e0faec7d89647ff9dcf8ba7f936a5a9313

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Citations in the corpus

Depleting the signal: Analysis of chemotaxis‐consumption models—A survey (2023, result): “9) when here the key ingredient φ is sufficiently regular and strictly positive on [0,∞) ([67], [75]; see also [72], [69] for precedent existence results, and [86], [78], [120] as well as [77, 96] for findings on some close relatives), a statement similar to those in (5.”
Refined regularity analysis for a Keller-Segel-consumption system involving signal-dependent motilities (2022, methodology): “…in (2) even enforces certain global very weak solutions, known to exist for all initial data from ( C 0 ((cid:4))) (cid:5) × L ∞ ((cid:4)) [22]; cf. also [23], to approach the semi-trivial equilibrium (cid:5) u 0 , 0 ) in the topology of ( W 1,2 ((cid:4))) (cid:5) × L ∞ ((cid:4)) in the…”