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 $\mathbb{R}^n$ with $n\geqslant 1$ and suitably regular initial data, where φ is assumed to reflect algebraic type cross-degeneracies by sharing essential features with $0\leqslant \xi\mapsto \xi^\alpha$ for some $\alpha\geqslant 1$. 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 $u_\infty$ and some null set $N_\star\subset (0,\infty)$, 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 $(0,\infty)\setminus N_\star i t\to \infty$, where $\rho(\xi): = \frac{\xi^2}{(\xi+1)^2}$, $\xi\geqslant 0$. In the particular case when either $n\leqslant 2$ and $\alpha\geqslant 1$ is arbitrary, or $n\geqslant 1$ and $\alpha\in [1,2]$, 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 $(u_\star,0)$ of ($\star$) is stable with respect to a suitably chosen norm topology.