Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar

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}$$\begin{aligned} \left\{ \begin{array}{l} u_t = \nabla \cdot \big ( uv\nabla u\big ) + \ell uv, \\ v_t = \Delta v - uv, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$\end{document}is considered in a smoothly bounded convex domain Ω⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^n$$\end{document}, with n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 1$$\end{document} and ℓ≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \ge 0$$\end{document}. The first of the main results asserts that for nonnegative initial data (u0,v0)∈(L∞(Ω))2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u_0,v_0)\in (L^\infty (\Omega ))^2$$\end{document} with u0≢0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0\not \equiv 0$$\end{document}, v0≢0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_0\not \equiv 0$$\end{document} and v0∈W1,2(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{v_0}\in W^{1,2}(\Omega )$$\end{document}, there exists a global weak solution (u, v) which, inter alia, belongs to C0(Ω¯×(0,∞))×C2,1(Ω¯×(0,∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^0(\overline{\Omega }\times (0,\infty )) \times C^{2,1}(\overline{\Omega }\times (0,\infty ))$$\end{document} and satisfies supt>0‖u(·,t)‖Lp(Ω)<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sup _{t>0} \Vert u(\cdot ,t)\Vert _{L^p(\Omega )}<\infty $$\end{document} for all p∈[1,p0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,p_0)$$\end{document} with p0:=n(n-2)+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_0:=\frac{n}{(n-2)_+}$$\end{document}. It is next seen that for each of these solutions one can find u∞∈⋂p∈[1,p0)Lp(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\infty \in \bigcap _{p\in [1,p_0)} L^p(\Omega )$$\end{document} such that, within an appropriate topological setting, (u(·,t),v(·,t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u(\cdot ,t),v(\cdot ,t))$$\end{document} approaches the equilibrium (u∞,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u_\infty ,0)$$\end{document} in the large time limit. Finally, in the case n≤5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\le 5$$\end{document} a result ensuring a certain stability property of any member in the uncountably large family of steady states (u0,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u_0,0)$$\end{document}, with arbitrary and suitably regular u0:Ω→[0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0:\Omega \rightarrow [0,\infty )$$\end{document}, is derived. This provides some rigorous evidence for the appropriateness of (⋆\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\star $$\end{document}) 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 (⋆\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\star $$\end{document}).

is considered in a smoothly bounded convex domain ⊂ R n , with n ≥ 1 and ≥ 0. The first of the main results asserts that for nonnegative initial data (u 0 , v 0 ) ∈ (L ∞ ( )) 2 with u 0 ≡ 0, v 0 ≡ 0 and √ v 0 ∈ W 1,2 ( ), there exists a global weak solution (u, v) which, inter alia, belongs to C 0 ( × (0, ∞)) × C 2,1 ( × (0, ∞)) and satisfies sup t>0 u(·, t) L p ( ) < ∞ for all p ∈ [1, p 0 ) with p 0 := n (n−2) + . It is next seen that for each of these solutions one can find u ∞ ∈ p∈[1, p 0 ) L p ( ) such that, within an appropriate topological setting, (u(·, t), v(·, t)) approaches the equilibrium (u ∞ , 0) in the large time limit. Finally, in the case n ≤ 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 : → [0, ∞), is derived. This provides some rigorous evidence for the appropriateness of ( ) 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 ( ).

Introduction
The question to which extent simple reaction-diffusion models can describe essential aspects of structure evolution in nature, and especially in living systems, has been stimulating considerable parts of the literature on parabolic problems over the past decades. Within the realm of bounded solutions, a dissipation-induced trend toward asymptotic equilibration has rigorously been confirmed to predominate in large classes of such systems ( [21], [22], [12], [14], [2] [9]). In line with this, substantial efforts have been focused on the identification of circumstances ensuring existence and favorable stability features of suitably structured steady states. Descriptions of structure-supporting properties on the basis of parabolic systems that exclusively involve reasonably regular ingredients seem accordingly constrained by natural limitations on richness of respectively relevant equilibria, despite a considerable collection of exceptions documented in the literature which assert the occurrence of impressively subtle stationary profiles especially near critical parameter settings (cf. [7], [8], [6], [4], [28], [24], [5] and also [25] for a very incomplete selection of findings in this regard).
The present study now addresses a modeling context which, despite a remarkable simpleness, seems characterized by an exceptionally large variety of evolutionary target states. Specifically, experimental findings reported in [11], [10] and [23] illustrate noticeably intricate facets in the collective behavior of populations of Bacillus subtilis when exposed to nutrientpoor environments; in particular, evolution into complex patterns, and even snowflake-like population distributions, appears as a generic feature rather than an exceptional event in such frameworks. Following experimentally gained indications for certain limitations of bacterial motility near regions of small nutrient concentrations, as a mathematical description for such processes the authors in [15] (cf. also [19] and [26]) propose the parabolic system for the population density u = u(x, t) and the food resource distribution v = v(x, t).
Forming the apparently most prominent ingredient herein, the diffusion operator in the first equation does not only degenerate in a standard porous medium-like manner at small values of u, but moreover accounts for said motility restrictions by containing a second degeneracy that emerges as soon as v approaches the level zero, in view of the second equation clearly expected to dominate at least on large time scales.
This latter peculiarity evidently brings about significant challenges beyond those mastered by classical approaches to degenerate diffusion problems of porous medium type ( [31], [2]), and also beyond those going along with the combination of two diffusion degeneracies due to the inclusion of diffusion rates that vanish both at small densities and at small gradient sizes of the diffusing quantity. In contrast to the situation in the latter class of doubly degenerate diffusion problems, meanwhile also fairly well-explored ( [13], [16], [29], [30]), especially the presence of a cross-degeneracy marks a considerable distinctiveness in (1.1). Accordingly, already at the stage of questions from basic existence theory the available knowledge so far seems limited to a statement on global solvability in a one-dimensional version ( [33]); built on a certain rather fragile energy structure, however, the corresponding arguments in [33] quite strongly rely not only on the presence of an additional cross-diffusive summand in the respective first equation and on strict positivity of , but beyond this also on the integrability assumption ln u(·, 0) > −∞ which especially rules out an analysis of solutions emanating from compactly supported initial data.
Main results. The present manuscript now attempts to develop an approach which is not only capable of launching a fundamental theory of general nonnegative solutions for the simple system (1.1) in multi-dimensional settings, but which moreover also allows for conclusions addressing qualitative questions in such cases, in particular with regard to aspects related to stabilization of structures. For convenience setting D = d = ϑ = 1 throughout the sequel, but keeping the parameter as an arbitrary nonnegative parameter in order to include the proliferation-free case = 0, we shall subsequently examine this in the framework of the initial-boundary value problem in a smoothly bounded domain ⊂ R n , with n ≥ 1, and with given initial data such that It is fairly evident that in this context, any expedient analysis of (1.2) needs to appropriately quantify the potentially weakened diffusive action therein. In our first step toward this, by merely relying on basic L 1 bounds for both components we shall use evolution properties of the sublinear functionals u q for q ∈ (0, 1) to gain a priori estimates for expressions of the form T 0 u q−1 v|∇u| 2 within this range of q (Lemma 2.3), which will in turn imply bounds, inter alia, for . The core part of our analysis will then be concerned with the derivation of higher L p bounds for u, where in the course of standard testing procedures a key requirement will consist in suitably exploiting the respective diffusion-induced contributions, as quantified through weighted expressions of type This will be achieved in Lemma 4.3, on the basis of the bounds known for the integral in (1.4), by means of two functional inequalities of the form valid with some C(K ) > 0 for arbitrary positive ϕ ∈ C 1 ( ) and ψ ∈ C 1 ( ) fulfilling throughout certain ranges of the parameters α, p, q and p (Lemma 4.1 and Lemma 4.2).
Within the framework of an appropriately regularized variant of (1.2), the a priori information thereby generated will firstly enable us to make sure that for any fixed initial data fulfilling (1.3), the problem (1.2) is solvable by functions inter alia belonging to C 0 ( × (0, ∞)) ∩ C 2,1 ( × (0, ∞)) and enjoying some time-independent bounds in L p ( ) × L ∞ ( ): such that u ≥ 0 and v > 0 in ×(0, ∞), and that (u, v) forms a global weak solution of (1.2) in the sense of Definition 2.1 below. Moreover, this solution has the additional boundedness property that sup By suitably taking into account respective dependences on time, our estimates gained in the context of (1.4), (1.5), (1.6) and (1.7) will, besides implying the above, already pave the way for our subsequent asymptotic analysis. By means of a duality-based argument, namely, the latter can be seen to ensure a decay feature of the form with C = C(u 0 , v 0 ) and some λ > 0, and with (W 1,∞ ( )) denoting the dual space of W 1,∞ ( ) (Lemma 5.1), and to thereby constitute the essential step toward our verification of the following statement on large time convergence of each individual among the trajectories obtained in Theorem 1.  11) and with some γ ∈ (0, 1), as t → ∞ we have (1.12) Finally, the third of our main results provides some rigorous analytical evidence for the suitability of (1.1) in the considered modeling context: Namely, by adequately tracing the dependence of the constant C = C(u 0 , v 0 ) in (1.10) on the initial data we can identify the following stability property enjoyed by actually any member of the uncountable family of function pairs (u 0 , 0), which for all suitably regular u 0 indeed form steady states of (1.
In particular, for the corresponding limit function u ∞ from Theorem 1.2 we then have

Approximation by positive classical solutions
The following solution concept to be pursued below seems to generalize notions of classical solvability in a manner fairly natural in contexts in which due to the considered diffusion degeneracy, no substantial first order information on the solution component u is expected.

Definition 2.1
Let n ≥ 1 and ⊂ R n be a bounded domain with smooth boundary, let ≥ 0, and assume that u 0 ∈ L 1 ( ) and v 0 ∈ L 1 ( ) are nonnegative. Then a pair of nonnegative functions In order to construct such solutions through some convenient approximation by classical solutions also in the presence of possibly discontinuous initial data, given u 0 and v 0 merely 1), and that u 0ε → u 0 and v 0ε → v 0 a.e. in as ε 0. (2.5) Then for each ε ∈ (0, 1), the regularized variant of (1.2) given by can be seen to actually inherit its initial non-degeneracy throughout evolution, and to thus allow for the following statement on global classical solvability by functions enjoying some rather expected basic features formally associated with (1.2): Lemma 2.2 Let n ≥ 1 and ⊂ R n be a bounded domain with smooth boundary, let ≥ 0, and assume (1.3) and (2.5). Then for each ε ∈ (0, 1), there exist functions Proof According to standard parabolic theory ( [1]), the regularity and positivity assumptions on u 0ε and v 0ε in (2.5) guarantee the existence of T max,ε ∈ (0, ∞] and positive functions (2.12) By nonnegativity of u ε v ε , the comparison principle ensures that this solution satisfies where the latter especially asserts that Another comparison argument therefore shows that writing and that hence, in particular, Again by comparison, we hence obtain that if we let Now assuming T max,ε to be finite for some ε ∈ (0, 1), we could combine (2.14) with (2.15) to infer that u ε v ε would then be bounded in × (0, T max,ε ), in view of standard parabolic regularity theory ( [17]) applied to the second equation in (2.6) implying that v ε would belong Since u ε v ε would then also be uniformly bounded from below by a positive constant in × (0, T max,ε ) by (2.13) and (2.16), and since thus the first equation in (2.6) would actually be uniformly parabolic in × (0, T max,ε ), we could employ the same token once again to infer from the first equation , again relying on uniform parabolicity of both equations in (2.6) we could then invoke parabolic Schauder theory ( [17]) to obtain , and that hence, in particular, Together with (2.13) and (2.16), in view of (2.12) this would contradict the hypothesis that T max,ε be finite. In consequence, (2.9) and (2.10) immediately result from (2.15) and (2.14), while (2.8) and (2.11) directly follow upon integrating in (2.6).
Throughout the remaining part of this manuscript, unless otherwise stated we shall tacitly assume that ⊂ R n is a bounded convex domain with smooth boundary, and that ≥ 0, and whenever (u 0 , v 0 ) and families ((u 0ε , v 0ε )) ε∈(0,1) fulfilling (1.3) and (2.5) have been fixed, by T max,ε and (u ε , v ε ) we shall exclusively mean the objects introduced in Lemma 2.2.

Fundamental first-order estimates. Basic decay properties of v
The following simple observation forms a fundament for the main part of our first step toward deducing further regularity properties of the solutions gained above, to be accomplished in Lemma 2.4 below on the basis of the estimate in (2.17). Apart from that, (2.17) will later on be utilized as a basic piece of information in our derivation of L p bounds for u ε (Lemma 4.3), and (2.18) as well as (2.19) will prove useful in the course of our large time analysis in Lemma 5.5 and Lemma 5.7.
and furthermore we have for all t > 0 and any ε ∈ (0, 1).
which firstly implies (2.19) as an immediate consequence of the restriction q ∈ (0, 1). After an integration in time, we moreover infer from (2.20) that thanks to the Hölder inequality, for all t > 0.
As u ε ≤ u 0ε + v 0ε for all t > 0 by (2.8), this establishes both (2.17) and (2.18). Now in the course of a standard testing procedure performed on the second equation in (2.6), an application of (2.17) to q := 1 2 enables us to control the respective interactiondriven contribution by means of conveniently decaying quantities. In particular, the following conclusion can therefore be drawn in such a way that besides providing first order regularity features, due to its temporally global nature it furthermore already includes some information on temporal decay in the second solution component; this latter aspect will be of essential importance to our argument asserting L p bounds for u ε (see Lemma 4.3). The following lemma is the only place in this manuscript in which convexity of is explicitly made use of.

Global existence
Besides on the boundedness properties documented in (2.9), (2.10) and (2.21), our construction of a solution to (1.2) through a limit procedure in Lemma 3.2 will rely on some further information on significantly enhanced regularity within finite time intervals bounded away from the initial instant. Due to its temporally local nature, the following statement in this regard, based on a combination of comparison arguments and standard parabolic theories, can apparently not be used in our large time analysis below, and hence does not trace dependencies of the obtained constants on the size of the initial data. and for all (x, t) ∈ × (τ, T ) and any ε ∈ (0, 1).
A weak solution with the additional properties announced in (1.8) can now be obtained by means of a straightforward extraction using the Arzelà-Ascoli theorem. n ≥ 1, and assume (1.3) and (2.5). Then there exist (ε j ) j∈N ⊂ (0, 1) and functions u and v fulfilling (1.8) such that ε j 0 as j → ∞, that u ≥ 0 and v > 0 in
Now the regularity requirements in (2.1) and (2.2) clearly result from the properties in (1.8) just asserted, and a verification of (2.3) and (2.4) can be achieved on the basis of (2.6) in a straightforward manner, using that due to (3.8) and (3.10) we have and that by (3.8) and (3.9),

Two functional inequalities
Up to this point, temporally global bounds which exclusively refer to u ε , without any presence of weight functions involving v ε as an expectedly decaying quantity such as in (2.18), seem limited to the L 1 boundedness feature in (2.8). A key step toward a more substantial description of the large time behavior in (1.2), and especially toward an exclusion of asymptotic Dirac-type mass accumulation, will now consist in an adequate control of the diffusion degeneracy in the first equation from (2.6), and particularly its part stemming from the presence of the factor v ε therein. In the framework of standard L p testing procedures, to be performed in Lemma 4.3, this specifically amounts to appropriately estimating integrals of the form from below, and to thereby control the temporal growth of u p ε , as potentially driven by the forcing term u ε v ε in (2.6).
Of crucial importance to our approach in this direction will be the following observation on how far expressions of the form in (4.1), when added to integrals of the type appearing in (2.22), dominate gradients of certain products involving u ε and v ε .
Then there exists C(α, p , K ) > 0 with the property that whenever ϕ ∈ C 1 ( ) and ψ ∈ C 1 ( ) are positive in and such that for any choice of q ∈ [0, 2α − 1] the inequality holds.
Proof We first observe that for any such ϕ and ψ, the pointwise estimate holds throughout with c 1 ≡ c 1 (α, p ) := (2α) 2 p p +2α−1 . Since p +2α−1 p > 1 thanks to our assumption that α > 1 2 , we may employ the Hölder inequality and rely on (4.2) to control the integral of the second summand on the right of (4.4) according to (4.5) Similarly, where using that the restrictions q ≥ 0 and q ≤ 2α − 1 warrant that 0 ≤ p (2α−q−1) 2α−1 ≤ p , we may invoke Young's inequality along with (4.2) to see that In conjunction with (4.6) and (4.5), this shows that (4.4) implies the inequality Combined with suitable Sobolev embedding properties, the latter entails the following class of interpolation inequalities appropriate for our purposes. Then for all K > 0 one can find C( p , K ) > 0 such that for any q ∈ [0, min{ p , 2 p n }] and for each ϕ ∈ C 1 ( ) and ψ ∈ C 1 ( ) fulfilling ϕ > 0 and ψ > 0 in as well as we have Proof We first consider the case n ≥ 2, in which W 1, 2n n+2 ( ) is continuously embedded into L 2 ( ), so that we can pick c 1 > 0 such that We may then apply Lemma 4.1 to α := 2 p +n 2n > 1 2 to infer the existence of c 2 ( p , K ) > 0 such that whenever ϕ and ψ are positive functions from C 1 ( ) which satisfy (4.8), for any Apart from that, we note that as a consequence of the Hölder inequality, any such pair (ϕ, ψ) satisfies because of (4.8). Together with (4.11) and (4.10), this already establishes (4.9) with in this case, for the restriction q ∈ [0, 2 p n ] clearly warrants that q ≤ 2α − 1.

Global L p estimates for u. Proof of theorem 1.1
Having Lemma 4.2 at hand, we are now prepared for our derivation of the following statement on L p boundedness of u ε , yet conditional in presupposing the existence of corresponding bounds in L p with some p ≥ 1, by means of the announced testing-based argument.
In particular, the latter completes our reasoning with regard to solvability and global L p regularity in (1.2): Proof of Theorem 1. 1 The part concerning existence and regularity has been completely covered by Lemma 3.2. The additional boundedness feature in (1.9) immediately follows from (4.19) when combined with (3.8).
Remark For initial data enjoying regularity and positivity features beyond those in (1.3), the existence result from Theorem 1.1 can be supplemented by a corresponding uniqueness statement by a straightforward combination of the reasoning from Lemma 2.2 with the standard theory developed in [1]: Indeed, if beyond (1.3) it was required that both u 0 and v 0 belong to q>n W 1,q ( ) and satisfy u 0 > 0 and v 0 > 0 in , then (1.2) could actually be seen to admit a global classical solution which is unique in the class of functions fulfilling {u, v} ⊂ q>n C 0 ([0, ∞); W 1,q ( ))∩C 2,1 ( ×(0, ∞)). Under the present mild hypotheses in (1.3) on the initial data, however, we do not expect solutions to be uniquely determined by the requirements in Definition 2.1, nor by the additional regularity features obtained in Lemma 3.2 which are yet fairly poor near the initial instant; we cannot even rule out the possibility that different choices of approximate initial data in (2.5) may lead to different limits.

Large time convergence of u in dual Sobolev spaces when n ≤ 5
Fortunately, in all physically relevant space dimensions the respective restrictions on p in Corollary 4.4 are mild enough so as to allow for the following conclusion on large time decay of u εt on the basis of (4.20), (4.21) and, again, the basic integrability property from (2.11).
λ for all ε ∈ (0, 1). As (5.1) involves L 1 norms with respect to the time variable only, in order to avoid any discussion of possibly measure-valued parts appearing in the time derivatives u t of the corresponding limits we prefer to formulate a conclusion of Lemma 5.1 for u which is sufficient for our subsequent reasoning in a version including temporal BV norms only, and hence exclusively involving the zero-order expression u itself: Corollary 5.2 Let n ≤ 5 and K > 0, let λ = λ(n) and C(K ) be as in Lemma 5.1, and assume (1.3), (2.5) and (1.13). Then for any where we have set u(·, 0) := u 0 .
When considering individual trajectories, we may here yet ignore any of the information about dependencies on initial data enclosed in (5.1), and to thereby obtain the following as a particular consequence. for all T > 0 and ε ∈ (0, 1). Since here Proof This can be concluded from Lemma 5.5 in much the same manner as Corollary 5.3 was derived from Lemma 5.1.

L 1 decay of v
By now making use of the lower bound for sublinear L q quasi-norms from Lemma 2.3, irrespective of the spatial dimension we can derive the following decay property of v from (2.11) and (2.21). Proof We fix any q ∈ (0, 1) such that 4q 3 ≤ 1, and from (2.8) we then obtain c 1 = c 1 (u 0 , v 0 ) > 0 such that for all t > 0 and ε ∈ (0, 1). (5.14) We moreover employ a Poincaré inequality to pick c 2 > 0 fulfilling ≤ η 2 + η 2 = η for all t > t 0 .
As η > 0 was arbitrary, this establishes the claim.

Proofs of theorems 1.2 and 1.3
Large time convergence in the claimed topological settings can now be obtained from Corollary 5.3, Corollary 5.6 and Lemma 5.7 by suitable interpolation using the boundedness features asserted by Theorem 1.1.
Proof of Theorem 1.2 As sup t>0 v(·, t) L ∞ ( ) is finite according to Theorem 1.1, for n ≥ 4 the convergence statement in (1.11) can be derived from Lemma 5.7 by a simple interpolation based on the Hölder inequality. In the case n ≤ 3, we once again use that thanks to (2.21) and Lemma 3.2, the integral ∞ 0 |∇v| 4 is finite, so that with some c 1 > 0 and some (t k ) k∈N ⊂ (0, ∞) fulfilling t k → ∞ as k → ∞ we have v(·, t k ) W 1,4 ( ) ≤ c 1 for all k ∈ N.
Since for any such n the Gagliardo-Nirenberg inequality provides c 2 > 0 fulfilling and that thus (1.11) also holds in this case, because being a classical solution of its respective sub-problem of (1.2) in × [1, ∞) by Theorem 1.1, thanks to the maximum principle the function v has the property that 0 < t → v(·, t) L ∞ ( ) is nonincreasing.
To deduce (1.12), we only need to combine the outcomes of Corollary 5.3 and Corollary 5.6 with the observation that for each κ > 0 and any p ∈ (1, n (n−2) + ), the family (u κ (·, t)) t>0 is bounded in L p κ ( ) by (1.9), and hence relatively compact with respect to the weak topology in this space.
Our main result on stability of arbitrary equilibria in (1.2) has actually been covered by Corollary 5.4 in its essence: Proof of Theorem 1.3 It is sufficient to apply Corollary 5.4, and to note that with (ε j ) j∈N taken from Lemma 3.
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