A quantitative strong parabolic maximum principle and application to a taxis-type migration–consumption model involving signal-dependent degenerate diffusion

By Michael Winkler. Annales de l'Institut Henri Poincare. Analyse non linéar, 2022

49 citations recorded by the source corpus.

The taxis-type migration-consumption model accounting for signal-dependent motilities, as given by ( u t = ∆ ( ) is considered for suitably smooth functions φ : [0 , ∞ ) → R which are such that φ > 0 on (0 , ∞ ), but that in addition φ (0) = 0 with φ ′ (0) > 0. In order to appropriately cope with the diﬀusion degeneracies thereby included, this study separately examines the Neumann problem for the linear equation and establishes a statement on how pointwise positive lower bounds for nonnegative solutions depend on the supremum and the mass of the initial data, and on integrability features of a and b . This is thereafter used as a key tool in the derivation of a result on global existence of solutions to ( ⋆ ), smooth and classical for positive times, under the mere assumption that the suitably regular initial data be nonnegative in both components. Apart from that, these solutions are seen to stabilize toward some equilibrium, and as a qualitative eﬀect genuinely due to degeneracy in diﬀusion, a criterion on initial smallness of the second component is identiﬁed as suﬃcient for this limit state to be spatially nonconstant.

Subfields: Modeling and Simulation, Molecular Biology, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/8e9e8424c524915ece5f94e971ec4287c9cadac4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Modeling and Simulation

In Molecular Biology

In Public Health, Environmental and Occupational Health

Citations in the corpus

Refined regularity analysis for a Keller-Segel-consumption system involving signal-dependent motilities (2022, background): “…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 large time limit when n ≤ 3 (see [22] and also Lemma 2.1 below); nontrivial large time dynamics in (2) seems possible only in degenerate cases in which φ( 0 ) = 0 [24,25].”