Small-signal solutions of a two-dimensional doubly degenerate taxis system modeling bacterial motion in nutrient-poor environments. (February 2022)

• Record Type: Journal Article
• Authors: Winkler, Michael
• Is Part Of: Nonlinear analysis. Volume 63(2022)
• Publication Date: 2022-02
• Subjects: Chemotaxis -- Degenerate diffusion -- A priori estimate -- Functional inequality; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2021.103407
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: The doubly degenerate nutrient taxis model u t = ∇ ⋅ ( u v ∇ u ) − ∇ ⋅ ( u 2 v ∇ v ) + ℓ u v, x ∈ Ω, t > 0, v t = Δ v − u v, x ∈ Ω, t > 0, is considered in smoothly bounded convex subdomains of the plane, with ℓ ≥ 0 . It is shown that for any p > 2 and each fixed nonnegative u 0 ∈ W 1, ∞ ( Ω ), a smallness condition exclusively involving v 0 can be identified as sufficient to ensure that an associated no-flux type initial–boundary value problem with ( u, v ) | t = 0 = ( u 0, v 0 ) admits a global weak solution satisfying ess sup t > 0 ‖ u ( ⋅, t ) ‖ L p ( Ω ) < ∞ . The proof relies on the use of an apparently novel class of functional inequalities which provide estimates from below for certain Dirichlet integrals involving possibly degenerate weight functions.