Small-signal solutions of a two-dimensional doubly degenerate taxis system modeling bacterial motion in nutrient-poor environments

The doubly degenerate nutrient taxis model {ut = del . (uvVu) - del . (u(2)v del v) + luv, x is an element of Omega, t > 0, v(t) = Delta v - Delta v, x E ohm, 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) is an element of W-1,W-infinity(Omega), a smallness condition exclusively involving v0 can be identified as sufficient to ensure that an associated no-flux type initial-boundary value problem with (u, v)|t=0 = (u0, v0) admits a global weak solution satisfying ess sup(t>0) parallel to u(center dot, t)parallel to L-p(Omega) < infinity. 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. (C) 2021 Elsevier Ltd. All rights reserved.