Global generalized solvability in a strongly degenerate taxis-type parabolic system modeling migration–consumption interaction

The parabolic problem ut=Δ(uϕ(v)),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}$$\begin{aligned} \left\{ \begin{array}{l} u_t=\Delta \big (u\phi (v)\big ), \\ v_t=\Delta v-uv, \end{array} \right. \end{aligned}$$\end{document}is considered in smoothly bounded subdomains of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document} with arbitrary 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}. Under the assumptions that ϕ∈C0([0,∞))∩C3((0,∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi \in C^0([0,\infty )) \cap C^3((0,\infty ))$$\end{document} is positive on (0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,\infty )$$\end{document} and satisfies lim infξ↘0ϕ(ξ)ξα>0andlim supξ↘0{ξβ|ϕ′(ξ)|}<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \liminf _{\xi \searrow 0} \frac{\phi (\xi )}{\xi ^\alpha }>0 \quad {\text{ and }} \quad \limsup _{\xi \searrow 0} \big \{ \xi ^\beta |\phi '(\xi )| \big \}<\infty \end{aligned}$$\end{document}with some α>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document} and β>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta >0$$\end{document}, for all reasonably regular initial data an associated no-flux type initial-boundary value problem is shown to admit a global solution in an appropriately generalized sense. This extends previously developed solution theories on problems of this form, which either concentrated on non-degenerate or weakly degenerate cases corresponding to the choices α=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =0$$\end{document} and α∈(0,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,2)$$\end{document}, or were restricted to low-dimensional settings by requiring that n≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\le 2$$\end{document}.


Introduction
This study is concerned with the problem x ∈ Ω, t > 0, ∇ uφ(v) · ν = ∇v · ν = 0, x∈ ∂Ω, t > 0, u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), x ∈ Ω, (1.1) which arises in the macroscopic modeling of migration processes involving a directing signal that is consumed upon contact, as typically found in nutrient-oriented movement such as in populations of aerobic bacteria, for instance. Forming a subclass of more general Keller-Segel-type chemotaxis systems in their consumptive specification [15,16], (1.1) postulates a precise link between the effective rates of diffusive and cross-diffusive migration, as implicitly entering the first equation therein through the summands ∇ · (φ(v)∇u) and ∇ · (uφ (v)∇v). Thereby particularly adapted to situations of chemotactic movement based on local sensing [3,7,16,22], the PDE system in (1.1) can be viewed as the consumption-related counterpart of corresponding chemotaxis-production variants that have been extensively studied in recent literature [1,3,10,11,27]; cf. also ([8,13,14,23-25,30,35] for some farther relatives); inter alia, both simulation-based and rigorous evidence indicates a considerable ability of such model classes to support spatial structures [4,5,9,11]. 32 Page 2 of 20 M. Winkler ZAMP Contrary to this, (1.1) with its enhanced dissipative action due to signal absorption seems hardly capable of generating any patterns in non-degenerate cases when the respective key ingredient φ is strictly positive. Indeed, in [19] it has been seen that whenever n ≥ 1, Ω ⊂ R n is a smoothly bounded domain and φ ∈ C 0 ([0, ∞)) is such that φ > 0 throughout [0, ∞), even for initial data merely belonging to (C 0 (Ω)) × L ∞ (Ω) a global very weak solution can always be found, and that if additionally n ≤ 3 and φ ∈ C 1 ([0, ∞)), then this solution approaches a semi-trivial homogeneous steady state in the large time limit (cf. also [20] for further regularity features of these solutions, and [21] for a precedent asserting global existence of classical small-data solutions). In the presence of certain diffusion degeneracies, however, (1.1) has been shown to facilitate patterning in small-signal settings, as experimentally observed in contexts of bacterial populations migrating in nutrient-poor environments [12,18,26]: If φ is suitably regular and such that then in one-and two-dimensional domains Ω, (1.1) is known to admit global weak solutions (u, v) which are even smooth and classical for t > 0, and for which there exists u ∞ ∈ p∈ [1,∞) if v 0 is appropriately small [32]. One element of motivation for the present study now consists in the question whether structure-supporting features of this flavor can be viewed as a widely encountered concomitant of migration inhibition in more general classes of diffusion degeneracies in (1.1), or whether the requirement (1.2) on essentially linear behavior of φ in fact forms a necessary prerequisite going beyond a purely technical restriction; in this context it may be noted that the apparently simplest choice φ ≡ id consistent with (1.2) would indeed lead to a precise coincidence of both nonlinearities in (1.1).
In the presence of more general degeneracies than those fulfilling (1.2), however, efficiently investigating (1.1) evidently amounts to appropriately coping with accordingly reduced a priori knowledge on regularity. In line with this, even at the stage of global solvability theories the available literature seems to have concentrated on particular scenarios sufficiently close to the low-dimensional setting involving (1.2). Specifically, under technical assumptions requiring that φ suitably generalizes the algebraic prototype given by for small values of ξ ≥ 0, some global very weak solutions to (1.1), mass-preserving but not necessarily bounded in their first component, could be constructed when yet n ≤ 2 but α > 0 is arbitrary [33], or when n ≥ 1 and α ∈ (0, 2] [34]. Main results: Global solvability in strongly degenerate higher-dimensional settings The intention of this manuscript is to develop an approach toward an expedient analysis of (1.1) which does not only allow for the inclusion of actually any such algebraic-type degeneracy, but which apart from that is widely independent of the spatial dimension and especially applicable also in the case n = 3 that seems to have nowhere been addressed so far. Our focus will here be on the design of a basic theory of global solvability, and hence on the design of methods capable of appropriately coping with the circumstance that in distinct contrast to related situations of scalar nonlinear parabolic problems of porous medium type [2,29], the derivation of regularity features despite diffusion-inhibiting mechanisms of cross-degenerate-and hence intricately nonlocal-character such as in (1.1) appears to require a simultaneous consideration of both sub-problems therein throughout core parts. Accordingly, the core of our analysis will be concerned with the examination of certain quasi-energy properties enjoyed by functionals of the form Global generalized solvability in a strongly degenerate taxis Page 3 of 20 32 along trajectories, within suitable ranges of the free parameters > 0 and m > 1 chosen in dependence of the prescribed system ingredient α > 0 that determines the behavior of φ near vanishing signal concentrations in the style of (1.3). A correspondingly obtained a priori estimate of the form (1.5) to be established for solutions to the approximate variants (2.1) of (1.1) in Lemma 3.4 whenever > 0 is arbitrary and m > 1 is adequately large, will firstly form a basis for the derivation of a suitable regularity feature of the associated time derivatives ∂ t (u + 1) − v m in Lemma 3.5. Secondly, despite the renormalization-induced low level of regularity level therein, these estimates can be seen to be sufficient for the construction of a limit object (Lemma 4.1), where of great importance will be the observation that as a consequence of a basic integral bound for u 2 φ(v) resulting from a standard duality argument, the nonlinearities from the respective second equations in (2.1) additionally enjoy a strong L 1 approximation property (Lemma 2.2, Lemma 2.3 and (4.6)). As thus the obtained candidate can be shown to indeed satisfy the second sub-problem of (1.1) in a standard weak sense (Lemma 4.1), by means of an essentially well-known argument it follows that the corresponding signal gradients actually exhibit a certain strong L 2 convergence property (Lemma 4.2). Along with associated weak L 2 approximation features of the gradients controlled by (1.5), this will imply that the gained candidate indeed also solves the first subproblem from (1.1) in an appropriately generalized sense (see  (1.6) and that there exist α > 0 and β > 0 such that as well as Then given any initial data (u 0 , v 0 ) such that

Basic properties of approximate solutions
Following [34], let us approximate (1.1) by considering the regularized variants thereof given by for ε ∈ (0, 1), where non-degeneracy of all these problems is ensured by the choice By copying known arguments, we can therefore verify the following statement on global existence of classical solutions to (2.1) which enjoy certain elementary regularity features.
Lemma 2.1. Let n ≥ 1 and Ω ⊂ R n be a bounded domain with smooth boundary, and assume (1.6) and (1.9). Then for each ε ∈ (0, 1) there exist and Proof. The statements concerning global solvability, mass conservation and the monotonicity feature in (2.5) can be confirmed by trivially extending the corresponding results detailed for the case n ≤ 2 in [33, Lemma 4.1] to domains of arbitrary dimension. The inequalities in (2.7), (2.6) and (2.8) can thereupon be derived from the identities Global generalized solvability in a strongly degenerate taxis Page 5 of 20 32 as well as as readily resulting for all t > 0 and ε ∈ (0, 1) from the second equation in (2.1), and estimating by means of (2.4).
Without further explicit mentioning, throughout the sequel we let the smoothly bounded domain Ω ⊂ R n and the initial data (u 0 , v 0 ) fulfilling (1.9) be fixed, and whenever a function φ fulfilling (1.6) has been chosen, we take ((u ε , v ε )) ε∈(0,1) be as accordingly provided by Lemma 2.1. By means of a method going back already to [27], thanks to the favorable structure of the migration operator in the first equation from (2.1) these solutions enjoy an additional fundamental regularity property independent from those obtained in Lemma 2.1.
Proof. This can be seen by copying the argument from [33, Lemma 3.1].
When combined with the L 1 boundedness feature contained in (2.4), this entails uniform integrability, though not of the first solution components alone, but after all for the products u ε v ε relevant to the absorptive reaction term in the second equation from (2.1): (2.10) Proof. Using that (α − 1)p ≤ α and hence α ≤ p p−1 , and that moreover p p−1 ≥ 2, we fix any α ≥ α such that α ≤ p p−1 and α ≥ 2. As the inequality α ≥ α together with (1.7) and (2.5) then ensures the existence of c 1 for all ε ∈ (0, 1), from Lemma 2.2 we thus infer that for any T > 0 we can find Now since α ≥ 2 > p, we may invoke Young's inequality to see that for all T > 0 and ε ∈ (0, 1), due to the fact that α ≤ p p−1 , upon another application of Young's inequality we find that thanks to A combination of (2.11) with (2.12) therefore leads to (2.10).
Apart from that, by simply using (2.7) along with (2.4) and (2.5) we can derive an estimate for the time derivatives of the signal concentrations with respect to some suitably mild norm.

A generalized notion of solvability
Now of fundamental importance for all our subsequent arguments will be the following identity describing the evolution of certain products which involve both solution components, and which are designed in such a way that from (2.5) their pointwise boundedness with respect to L ∞ norms, uniform with respect to ε, is a priori known.
Global generalized solvability in a strongly degenerate taxis Page 7 of 20 32 Proof. We use (2.1) and integrate by parts to compute ϕ for all t ∈ (0, T ) and ε ∈ (0, 1), from which (3.1) follows upon suitable rearrangement.

Corollary 3.2.
Proof. This can be seen by means of straightforward modification of (3.1) using that This observation can now be taken as a blueprint for our notion of generalized solvability in the original problem (1.1). In fact, in the spirit of a conceptual approach which shares essential features with the renormalization idea by Di Perna and Lions [6], and which in the context of taxis-type parabolic problems apparently goes back to [17] and [31], on the basis of (3.2) we require a solution of (1.1) to comply with the following.
be such that u ≥ 0 and v > 0 a.e. in Ω × (0, ∞). Then (u, v) will be called a global generalized solution and if there exist > 0 and m ≥ 1 such that and that for each nonnegative ϕ ∈ C ∞ 0 (Ω × [0, ∞)), the inequality Global generalized solvability in a strongly degenerate taxis Page 9 of 20 32 holds.

Remark 1.
By an essentially verbatim copy of the arguments from [17, Lemma 2.5], it can be seen that this concept indeed is consistent with that of classical solvability in the sense that if (u, v) ∈ ) is a global generalized solution of (1.1) in the sense of Definition 3.3, then (u, v) satisfies (1.1) also in the classical pointwise sense.

A priori estimates focused on the requirements of Definition 3.3
Thus having a clear goal in mind now, we next plan to derive a priori estimates for the quantities relevant to Definition 3.3, and especially to the crucial part (3.8) therein. For this purpose, we first go back to Lemma 3.1, this time applied to values of m slightly differing from those in Corollary 3.2, to address the gradients from (3.6). Then there exists C > 0 such that Proof. We let λ := 2 and μ := 2m − α, and then note that μ is positive by (3.9), and that thanks to An application of Lemma 3.1 to ϕ := 1 accordingly shows that d dt for all t > 0 and ε ∈ (0, 1). (3.12) Here by Young's inequality and (3.11), for all t > 0 and ε ∈ (0, 1). Since here 2m − 2α − 2β and 2m − 2α − 2 are both nonnegative by (3.9), again thanks to (3.11) |∇v ε | 2 for all t > 0 and ε ∈ (0, 1). (3.13) Apart from that, on the right-hand side of (3.12) we can once more utilize (3.11) to estimate for all t > 0 and ε ∈ (0, 1), (3.14) because again due to (3.9) From (3.12) we thus infer that thanks to the nonpositivity of the last summand therein, and thanks to (3.13) and (3.14), with |∇v ε | 2 for all T > 0 and ε ∈ (0, 1). (3.15) Observing that for all t > 0 and ε ∈ (0, 1), from (3.15) we conclude that (3.10) is a consequence of (2.7).
To finally establish our main result on global solvability in (1.1), it is sufficient to simply summarize. .
Funding Open Access funding enabled and organized by Projekt DEAL. The author acknowledges support of the Deutsche Forschungsgemeinschaft (Project No. 462888149).

Conflict of interest
The author declares that he has no conflict of interest.
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