Stabilization despite pervasive strong cross-degeneracies in a nonlinear diffusion model for migration–consumption interaction

Abstract

A no-flux initial-boundary value problem for ut=Δ(uϕ(v)),vt=Δv−uv,(⋆) is considered in smoothly bounded subdomains of with and suitably regular initial data, where φ is assumed to reflect algebraic type cross-degeneracies by sharing essential features with for some . Based on the discovery of a gradient structure acting at regularity levels mild enough to be consistent with degeneracy-driven limitations of smoothness information, in this general setting it is shown that with some measurable limit profile and some null set , a corresponding global generalized solution, known to exist according to recent literature, satisfies ρ(u(⋅,t))⇀⋆ρ(u∞)in L∞(Ω) and v(⋅,t)→0in Lp(Ω)for all p⩾1 as , where , . In the particular case when either and is arbitrary, or and , additional quantitative information on the deviation of trajectories from the initial data is derived. This is found to imply a lower estimate for the spatial oscillation of the respective first components throughout evolution, and moreover this is seen to entail that each of the uncountably many steady states of () is stable with respect to a suitably chosen norm topology.