A degenerate migration-consumption model in domains of arbitrary dimension

1 Introduction

Local sensing mechanisms are relevant to partially directed motion of cell motion ([1]–[5]). It their simplest form, macroscopic models for such processes describe the population density u = u(x, t) by parabolic equations of the form

([1], [2]), where in typical application situations, the cell motility coefficient a may depend on a chemical substance, represented through its concentration v = v(x, t) itself forming an unknown of the system, via various functional laws ([3], [6]).

In recent analytical literature, significant activity has been directed toward an understanding of resulting two-component parabolic models in cases in which the respective signal is produced by cells, and which thus, as in the classical Keller-Segel systems from [7], reflect taxis-mediated active communication between cells. Hence focusing on systems of the form

as well as on some parabolic-elliptic simplifications thereof, a considerable collection of studies has identified various conditions on the key ingredient ϕ as sufficient for global solvability and hence for suppression of finite-time blow-up (see [8], [9], [10]–[13], [14], [15] for an incomplete selection, and also [16], [17]–[19], [20], [21], [22] for some studies on variants accounting for sources and density-dependent diffusion mechanisms); on the other hand, some results detecting the occurrence of infinite-time blow-up in the particular case when ϕ(v) = e ^−v , v ≥ 0, indicate a certain reminiscence of Keller-Segel dynamics ([11]; cf. also [13]): Passing over from classical Keller-Segel-production systems to models of the form (1.2) may thus, depending on the choice of ϕ, delay but not entirely rule out unboundedness phenomena ([9]).

In comparison to the above, much less seems known for related systems addressing situations of local sensing in which the directing signal is consumed by individuals, and in which thus cells, in particular, are incapable of active communication. In fact, for corresponding migration-absorption models of the form

the literature so far appears to concentrate on non-degenerate cases determined by motilities which are strictly positive on [0, ∞), and which hence reflect non-degenerate diffusion: In such situations, the additional dissipative influence exerted by the absorptive reaction substantially facilitates global existence theories, in frameworks both of classical small-data and of generalized large-data solutions ([23], [24]); as strongly indicated by quite far-reaching findings on large time stabilization toward spatially homogeneous steady states, however, non-degenerate settings of this flavor seem unable to adequately capture any of the strongly structure-supporting features of collective movement observed in populations of aerobic bacteria ([25]).

This is in line with refined modeling approaches which, in order to particularly address such situations, suggest to explicitly account for reduction of bacterial motility in nutrient-poor environments ([25], [26]). Indeed, a recent result indicates that in sharp contrast to said case of positive ϕ, nontrivial long term dynamics may indeed occur in corresponding versions of (1.3) which accordingly include migration rates reflecting motility degeneracies at small signal concentrations: When ϕ is suitably smooth with

namely, an associated no-flux type boundary value problem for (1.3) in one- or two-dimensional domains has been found to admit some classical solutions (u, v) for which u approaches a nonconstant profile in the large time limit ([27]). However, the question whether such types of behavior are restricted to such special settings, or rather constitute a characteristic feature of degenerate motilities in (1.3) within a more general framework, appears to be open up to now; in particular, the only precedent we are aware of which addresses somewhat stronger degeneracies, in fact covering any decay behavior of ϕ near the origin which is of essentially algebraic type, is still limited to domains in R n with n ≤ 2 ([28]).

Main results. The present manuscript attempts to design an analytical approach for (1.3) which does not only allow for the inclusion of motility degeneracies more general than those determined by (1.4), but which moreover does not rely on assumptions on low dimensionality. Due to challenges which in comparison to those encountered in the setup from (1.4) seem considerably increased, we will focus here on issues from basic solution and regularity theories, leaving more detailed qualitative investigation for future research.

Specifically, we shall consider the initial-boundary value problem

in n-dimensional smoothly bounded convex domains Ω, under the assumption that near the origin, ϕ suitably generalizes the prototype given by

with certain α > 0 and ξ ₀ > 0.

Our first step will concentrate on the case α ∈ (0, 1), in which on the one hand a comparatively mild degeneracy retains some strength of diffusive smoothing, but for which on the other the corresponding cross-diffusive action is considerably singular in regions where v is small. In the context of the identity

formally determining the evolution of the associated logarithmic entropy, the latter becomes manifest in a singular factor ϕ′(v) appearing in the rightmost integral, and a straightforward estimation thereof in terms of the dissipated quantity on the left thus seems not expedient. Forming a key observation in this regard, it will turn out that by linearly combining (1.7) with a corresponding identity describing the evolution of an appropriate bounded quantity, this expression can be suitably diminished in strength, and hence become conveniently controllable by respective diffusive contributions. Indeed, in Section 3 we shall see that for appropriately chosen a > 0, an inequality of the form

holds for solutions to certain regularized variants of (1.5) (see (2.5) and Lemma 3.4). In conjunction with some basic regularity features, originating from a standard duality-based reasoning and providing bounds for both summands on the right of (1.8) (Lemma 2.4, Lemma 2.7 and Corollary 3.1), this will lead to a priori information sufficient for the derivation of a result on global existence of weak solutions with locally bounded logarithmic entropies.

More precisely, the first of our main results can be stated as follows.

Next concerned with stronger degeneracies, we will need to appropriately cope with an apparent lack of structural features comparable to that from (1.8) in the case when α ≥ 1. We shall alternatively build our analysis in this respect on the observation that as long as α ∈ (1, 2), a favorable entropy-like evolution property of Dirichlet integrals involving mildly singular weights will lead to an inequality of the form

(Lemma 4.3). Along with fairly standard extensions thereof to the borderline cases α = 1 and α = 2 (Corollary 4.4), this can be combined with an again duality-based control of ∫ 0 T ∫ Ω u 2 v α which now can even be achieved with bounds uniform with respect to T thanks to the fact that uv ^α is esentially dominated by the quantity uv known to belong to L ¹(Ω × (0, ∞)) according to the second equation in (1.5) (Lemma 4.1 and (2.10)). In Lemma 4.5, we thereby see that whenever α ∈ [1, 2], a fairly straightforward estimation of the right-hand side in (1.7) becomes possible so as to ensure bounds which are now even independent of time.

In conclusion, this will enable us to derive the second of our main results, asserting global solvability and temporally uniform bounds in the presence of such superlinear degeneracies, and in arbitrarily high-dimensional settings, in the following sense.

We remark that the theory developed here forms the basis of the refined qualitative analysis undertaken in [29]; further information on large time stabilization has been obtained in [30] and in [27].

2 Approximation and some basic regularity features

The solution concept to be considered below appears to be quite in line with standard notions of generalized solvability in second order parabolic problems, particularly involving one-step integration by parts only.

In order to achieve a convenient approximation of (1.5), let us regularize not only the diffusive contribution to the first equation, but also the reaction part in the second. In fact, this will ensure that for each ɛ ∈ (0, 1), the problem

with

is globally solvable in the classical sense:

Throughout the sequel, we shall henceforth fix a smoothly bounded Ω ⊂ R n as well as initial data (u ₀, v ₀) fulfilling (1.13), and whenever a function ϕ satisfying (1.9) is given, we shall let ( ( u ε , v ε ) ) ε ∈ ( 0,1 ) denote the family of solutions to (2.5), (2.6) accordingly obtained in Lemma 2.2.

A first common regularity feature of these solutions will result from a duality-based argument based on the following observation which in its essence goes back to [15] already. Here and below, for φ ∈ L ¹(Ω) we abbreviate its average according to φ ̄ ≔ 1 | Ω | ∫ Ω φ .

In a general setting compatible with the assumptions both of Theorem 1.1 and Theorem 1.2, this can be seen to imply a first a priori estimate beyond those from Lemma 2.2. We announce already here, however, that in the context of nonlinearities ϕ which grow at most linearly near the origin, Lemma 4.1 will provide a significant refinement which will form the origin for the time-independent bounds claimed in Theorem 1.2.

Playing a key role in our reasoning, the standard logarithmic entropy can be described with respect to a very basic evolution feature as follows.

Thus led to provide appropriate control over the expression on the right of (2.16) and especially the taxis gradients ∇v _ɛ , possibly with singular weights originating from potentially unbounded factors ϕ ε ′ ( v ε ) , we first perform a very standard testing procedure to obtain a general statement on a basic regularity property, which at this stage is yet conditional by relying on a square integrability feature of ( u ε v ε ) ε ∈ ( 0,1 ) .

As a crucial preparation for our analysis of (2.16) in the context of both Theorem 1.1 and the particular subcase α = 1 of Theorem 1.2, we note that if integral bounds even for u ε 2 v ε can be drawn on, then the taxis gradient can be controlled even when weighted in a more singular manner than in (2.18). This can be confirmed in the course of another fairly well-established variational reasoning:

The following analysis of the products u ε v ε 2 will provide a handy path toward the derivation of a favorable compactness feature which, thanks to the positivity feature of the v _ɛ encrypted in (2.11), will form the source for pointwise a.e. convergence of ( u ε ) ε ∈ ( 0,1 ) along a suitable subsequence (cf. Lemma 3.6).