A no-flux initial-boundary value problem for the doubly degenrate parabolic system ut=∇·(uv∇u)+ℓuv,vt=Δv-uv,(⋆)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} u_t = \ abla \\cdot \\big ( uv\ abla u\\big ) + \\ell uv, \\\\ v_t = \\Delta v - uv, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$\\end{document}is considered in a smoothly bounded convex domain Omega subset mathbb {R}^n, with nge 1 and ell ge 0. The first of the main results asserts that for nonnegative initial data (u_0,v_0)in (L^infty (Omega ))^2 with u_0not equiv 0, v_0not equiv 0 and sqrt{v_0}in W^{1,2}(Omega ), there exists a global weak solution (u, v) which, inter alia, belongs to C^0(overline{Omega }times (0,infty )) times C^{2,1}(overline{Omega }times (0,infty )) and satisfies sup _{t>0} Vert u(cdot ,t)Vert _{L^p(Omega )}<infty for all pin [1,p_0) with p_0:=frac{n}{(n-2)_+}. It is next seen that for each of these solutions one can find u_infty in bigcap _{pin [1,p_0)} L^p(Omega ) such that, within an appropriate topological setting, (u(cdot ,t),v(cdot ,t)) approaches the equilibrium (u_infty ,0) in the large time limit. Finally, in the case nle 5 a result ensuring a certain stability property of any member in the uncountably large family of steady states (u_0,0), with arbitrary and suitably regular u_0:Omega rightarrow [0,infty ), is derived. This provides some rigorous evidence for the appropriateness of (star ) to model the emergence of a strikingly large variety of stable structures observed in experiments on bacterial motion in nutrient-poor environments. Essential parts of the analysis rely on the use of an apparently novel class of functional inequalities to suitably cope with the doubly degenerate diffusion mechanism in (star ).
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