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