The role of superlinear damping in the construction of solutions to drift-di usion problems with initial data in L 1

is considered for su ciently regularmatrix-valued A, vector-valued b and real valued g, andwith f representing superlinear absorption in generalizing the prototypical choice given by f (·, ·, s) = sα with α > 1. Problems of this form arise in a naturalmanner as sub-problems in several applications such as cross-di usion systems either of Keller-Segel or of Shigesada-Kawasaki-Teramoto type in mathematical biology, and accordingly a natural space for initial data appears to be L1(Ω).


Introduction
A common feature of numerous evolution equations stemming from population models in mathematical biology is the appearance of superlinear degradation terms. In applications typically interpreted, depending on the respective modeling approach, as accounting for diminution due to competition, or as more generally representing abilities of systems to spontaneously prevent overcrowding, such expressions typically arise in the form of algebraic zero-order absorption terms. In the simplest case combined merely with di usion and thus resulting in semilinear heat equations such as degradation mechanisms of this type usually provide additional dissipation resulting in accordingly enhanced relaxation features. A favorable mathematical e ect thereof is that despite their nonlinear character, such absorptive nonlinearities do not essentially counteract existence theories; in fact, su ciently elaborate analysis shows that the superlinear damping in (1.1) can be used to even expand the well-known solution theory for the heat equation so as to construct solutions even for very singular initial data with regularity properties far below integrability (see [4,28,29,43] and the detailed discussion in the latter, for instance).
That this situation may substantially change when such absorption interacts with further and possibly destabilizing mechanisms is indicated by ndings on extensions of (1.1) to systems involving cross-di usion, such as the logistic Keller-Segel system ( [15]) Indeed, the solution theories for both these systems are much less developed than that for e.g. (1.1), which may be viewed as partially re ecting a certain singularity-supporting potential of the respective transport processes therein; drastic caveats in this direction are provided by studies reporting the taxis-driven occurrence of large densities in several versions of (1.2) for α = ( [14,16,39,41]), and even detecting nite-time blow-up of some solutions to (1.2) in n-dimensional balls with n ≥ , for τ = and α ∈ ( , α (n)) with some α(n) ∈ ( , ), even for smooth initial data ( [42], cf. also [38]).
Apart from accordingly implied natural limitations, the construction of global solutions to both (1.2) and (1.3) in the literature has been confronted with signi cant additional and possibly technical challenges, and thus in successful cases been strongly relying on the presence of particular global dissipative features expressed in corresponding energy or at least quasi-energy inequalities. For instance, the discovery of an appropriate Lyapunov-like functional has given rise to a breakthrough in the existence theory, within suitably weak solution concepts, for (1.3) with widely arbitrary parameters therein ( [7]), thus complementing and extending results on global solvability in classes of smooth functions but under various types of more or less restrictive assumptions on the system ingredients ( [8,12,19,21,22]). Similarly, the use of certain quasi-energy structures in (1.2) has formed an essential fundament for the construction of global bounded solutions in suitable parameter regimes and in presence of su ciently regular initial data ( [23,30,36]).
Beyond the evident circumstance that such structures are commonly quite sensitive with respect to changes in the system ingredients, an apparent application-relevant restraint stems from the observation that a corresponding analysis usually requires the initial data to be regular enough so as to have the associated energy be nite at the initial instant. In the context of (1.3), this leads to the requirement, apparently underrun nowhere in the literature, that u := u| t= at least be an element of an L log L-type Zygmund class; as for (1.2), most works even assume continuity of the initial data. Up to one single exception addressing global existence of certain generalized solutions to (1.2) in the simple case τ = with α > − n , however, the literature does not provide any result on solvability in parabolic drift-di usion systems of the form (1.2) or (1.3), to say nothing of providing a generalizing or even unifying point of view, in situations when initial data are merely assumed to be integrable, and thus to comply with essentially minimal requirements meaningful in the context of applications in which u usually plays the role of a total population size.
Main objective: Construction of generalized solutions with initial data in L . Methodologically, the main challenges going along with the treatment of less regular initial data seem to be linked to the derivation of appropriate compactness properties of the respective superlinear reaction terms, thereby allowing for suitable limit procedures in conveniently designed approximate problems. Here we especially emphasize that due to the presence of additional drift-type mechanisms therein, the accessibility of cross-di usion systems like (1.2) and (1.3) to compactness-revealing techniques based on duality arguments, as recently developed to quite a comprehensive extent in frameworks of certain pure reaction-di usion systems generalizing (1.1) to corresponding multi-component problems ( [5,24]), seems very limited.
Accordingly, a common characteristic feature of virtually all precedent solution constructions for (1.2) and (1.3) consists in asserting equi-integrability properties of the nonlinearities in question by tracking the time evolution of convex functionals of the crucial unknown u, with u ln u consituting the most frequently seen representative. Due to the absorptive character of degradation, namely, the associated testing procedures, essentially involving increasing functions of u as test functions in the respective rst equations, yield favorably signed contributions that involve functionals of u with conveniently fast growth as u → ∞. Indeed, corresponding multiplication by ln u, e.g. in (1.2) resulting in space-time L estimates for u α ln u and hence implying suitable (equi-)integrability features of u α , has been at the core of various existence proofs in (1.2) as well as in several related taxis-type systems ( [17,27,44]); through their mere nature, however, such techniques seem restricted to cases in which, again, not only u but even some superlinear functional of u is integrable.
The purpose of the present work is to develop an apparently alternative approach toward the construction of generalized solutions, rstly mild enough with regard to the initial data so as to be applicable to data merely belonging to L , and secondly su ciently robust in not relying on fragile structures like entropies. We shall accordingly be concerned with a rather general class of systems involving superlinear degradation, possibly furthermore perturbed by drift terms, by subsequently considering the no-ux type parabolic problem where T ∈ ( , ∞] and Ω ⊂ R n is a bounded domain with smooth boundary. Here we assume throughout that the di usion operator generalizes the Laplacian in that with some positive constants k A and K A , that the drift coe cient satis es the crucial square integrability condition b ∈ L loc (Ω × [ , T); R n ), (1.6) that the nonlinear part of the reaction term, with some k f > , K f > and α > , and that moreover g ∈ L loc (Ω × [ , T)) (1.9) and u ∈ L (Ω) is nonnegative. (1.10) Main results I: Constructing very weak solutions without need for L compactness properties of u α .
In view of the above observations on precedent studies, our rst objective will consist in examining how far solutions can be obtained even despite possibly lacking estimates ensuring compactness features that allow for standard limit passages in classical weak formulations associated with (1.4). For this purpose, in a rst step we shall further develop an approach from [40] by resorting to a solution concept which in its most crucial part concentrates on the function ln(u + ) and merely requires this quantity to satisfy an integral inequality re ecting a certain supersolution property of ln(u + ) with respect to its parabolic problem formally corresponding to (1.4); along with a suitable additional mass control from above, this yields a concept which for smooth functions is indeed consistent with classical solvability. The main advantage of this relaxation consists in the circumstance that in comparison to standard notions of weak solvability, such as formulated e.g. in De nition 3.1 below, with respect to the decisive nonlinear parts this will here require signi cantly reduced integrability and compactness properties only, which we will see to indeed be available in quite a general framework.
More precisely, in this rst part we shall adapt a concept originally introduced in [40] for a particular chemotaxis problem, and later on extended to various relatives thereof (see e.g. [3,34]), in the following manner.
De nition 1.1. Let T ∈ ( , ∞], and suppose that (1.5), (1.6), (1.7), (1.9) and (1.10) hold with some k A > and K A > . Then a nonnegative function u ∈ L loc (Ω × [ , T)) will be called a very weak solution of (1.4)  Now to substantiate our approach toward solvability in the context of a convenient approximation to (1.4), let us further specify our setting by imposing the hypothesis, forming a standing assumption in this general part, that from whatever source we are given nonnegative classical solutions uε ∈ C (Ω × [ , T)) ∩ C , (Ω × ( , T)) to the regularized variants of (1.4) speci ed by (1.14) where ε ∈ (ε j ) j∈N with some sequence (ε j ) j∈N ⊂ ( , ) ful lling ε j ↘ as j → ∞. As for the ingredients herein, in line with the above we will assume that with positive constants k A , K A , k f , K F and s , without loss of generality coinciding with those introduced above, we have (1.18) and that the functions Finally, the initial data in (1.14) will be subject to the assumptions that The rst of our main results, to be achieved in Section 2, then asserts that these approximation properties, and especially the crucial L convergence requirement in (1.18), ensure solvability in the considered very weak framework, indeed assuming no more regularity of u than merely integrability: (1.6), (1.7), (1.8), (1.9) and (1.10) hold for some T ∈ ( , ∞], k A > , K A > , k f > , K f > , s > and α > , and for ε ∈ (ε j ) j∈N with some sequence (ε j ) j∈N ⊂ ( , ) such that ε j ↘ as j → ∞, assume that uε ∈ C (Ω×[ , T))∩C , (Ω×( , T)) is a classical solution of (1.14) with certain Aε , bε , gε and u ε satisfying (1.15), (1.16), (1.17), (1.18), (1.19), (1.20), (1.21) and (1.22). Then there exist a subsequence (ε j k ) k∈N and a very weak solution u of (1.4) in Main results II: Construction of genuine weak solutions by turning weak into strong L α convergence for su ciently regular b. The major step in our analysis thereafter consists in investigating how far despite the mentioned obstacles the solution gained above in fact solves (1.4) in the standard weak sense. In view of (1.8), this essentially amounts to identifying conditions under which the weak convergence statement in (1.24) can be turned into a corresponding strong compactness property, where in accordance with the above discussion, our ambition to avoid further regularity requirements on the initial data apparently reduces the availability of well-established techniques which in related situations have provided equi-integrability features of, say, some family (h j ) j∈N by deriving L bounds for (Ψ(h j )) j∈N with certain superlinearly growing Ψ : R → R ( [7,17,24,27]).
In our key step toward circumventing this, we will purely concentrate on the weak supersolution property satis ed by the limit function u due to Theorem 1.2, and the main challenge here will be to create an appropriate testing procedure in the corresponding integral inequality which allows for a rigorous justi cation of the mass evolution relation t Ω u α ε , and that hence f (·, ·, uε) → f (·, ·, u) in L (Ω × ( , t )), for suitably many t ∈ ( , T). We underline already here that developing (1.26) from the inequality (1.12) will go along with considerable e orts, especially due to the circumstance that (1.12) addresses ln(u + ) rather than u itself, and that according to the poor regularity information available for u, quite restrictive requirements for the corresponding test functions are in order.
It will turn out in Section 3, however, that under slightly sharpened assumptions on α and the integrability properties of b this can successfully be accomplished, thus leading to the following result. Application to logistic Keller-Segel systems. To indicate how the above general theory can be employed in the construction of solutions to concrete cross-di usion systems involving couplings to further quantities, in Sections 4 and 5 we will focus on the two examples (1.2) and (1.3) introduced above; in order to avoid to become too extensive here, we only mention that further applications to several models of biological relevance are possible, including chemotaxis-haptotaxis systems for tumor invasion or coupled chemotaxis-uid systems, for instance ( [2,6]).
Let us rstly consider the Neumann problem for the relative of (1.2) given by where with some k F > , K F > , s > and α > , and where u ∈ L (Ω) and v ∈ L (Ω) are nonnegative, with a particular representative constituted by the classical logistic Keller-Segel system with quadratic degradation, as given by for λ ∈ R and µ > . It is known from the literature that for initial data additionally satisfying u ∈ C (Ω) and v ∈ W ,∞ (Ω), the latter problem admits global classical solutions when either n ≤ and µ > is arbitrary ([23]), or n ≥ and µ > µ (λ, Ω) with some µ (λ, Ω) > ( [36]); for arbitrary values of µ > and suitably regular data, global weak solutions have been obtained in [17]. Analytic studies focusing on solvability issues in presence of smaller powers α in the degradation term F from (1.28) and (1.29) apparently go back to [35] where some global generalized solutions could be constructed for a parabolic-elliptic relative under the assumption that α > − n , with a recent extension to the fully parabolic case (1.28) for smooth initial data achieved in [33].
Now based on an application of Theorem 1.2, some considerable relaxation with regard to both the condition (1.31) and the initial regularity becomes possible, thus leading to a result on solvability in the fully parabolic problem (1.28) not only for initial data merely belonging to L × L , but apart from that also for a range of degradation parameters α apparently not addressed by any existence result in the literature so far: and that (u, v) forms a very weak solution of (1.28) in Ω × ( , ∞) in the sense that u is a very weak solution on (1.4)  for all φ ∈ C ∞ (Ω × [ , ∞)). This solution can be obtained as the limit of classical solutions (uε , vε) to (4.3) below in the sense that there exists (ε j ) j∈N ⊂ ( , ) such that ε j ↘ as j → ∞ and that uε → u and vε → v a.e. in Ω × ( , ∞) as ε = ε j ↘ .
Under slightly stronger assumptions on α and the initial regularity of v, yet retaining the mere requirement u ∈ L (Ω), we shall next derive from Theorem 1.   (1.37) and which form a weak solution of (1.28) in Ω × ( , ∞) in the sense that (1.34) holds and that u solves (1.4) with A, b, f and g as speci ed in Theorem 1.4; in particular, In the particular context of the system (1.30) with quadratic degradation, the latter implies the following.
Corollary 1.6. Let n = , λ ∈ R and µ > , and suppose that ≤ u ∈ L (Ω) and ≤ v ∈ W , (Ω). Then there exist nonnegative functions u and v on Ω × ( , ∞) such that for any T > we have and that (u, v) solves (1.30) in the weak sense speci ed in Theorem 1.5.
Application to a Shigesada-Kawasaki-Teramoto type system. Finally, we brie y address a speci c version of the comprehensive model (1.3), reducing the full complexity therein by resorting to a tridiagonal case in which cross-di usion enters only one of the equations. Up to the exceptional approach based on exploiting global entropies ( [7]), such simpli cations have been an essential prerequisite in most previous studies on global solvability in the context of (1.3), mainly in frameworks of smooth solutions for smooth initial data ( [8,12,19,21,22,32]).
Speci cally, we will focus on the system and rstly derive from Theorem 1.2 the following existence result for data in L × L ∞ .

Theorem 1.7.
Let Ω ⊂ R n be a bounded domain with smooth boundary, let d , d and µ be positive and a , a , µ , a and a be nonnegative, and let ≤ u ∈ L (Ω) and ≤ v ∈ L ∞ (Ω). Then one can nd nonnegative functions u and v on Ω × ( , ∞) such that for all T > , 40) and such that u is a very weak solution of (1.4) in Ω × ( , ∞) in the sense of De nition 1.
In order to identify this very weak solution as an actually weak solution by means of Theorem 1.3, we here only need to invest the additional hypothesis that v belong to W , (Ω). and which constitute a weak solution of (1.28) in Ω × ( , ∞) in that (1.40) holds for all φ ∈ C ∞ (Ω × [ , ∞)), and that u is a weak solution of (1.4) in the sense of De nition 3.1 with A, b, f and g as speci ed in Theorem 1.7; in particular, we have

Solvability despite lacking strong compactness. Proof of Theorem 1.2
In order to construct a very weak solution by means of a limit procedure involving supposedly given classical solutions of the regularized problems (1.14), let us assume throughout this section that ( Then a basic but important property can immediately be seen. Proof. Thanks to the no-ux boundary condition in (1.14), integrating the rst equation therein yields which directly leads to (2.1). As a consequence of (1.8), under an additional assumption on the positive part of gε, actually weaker than our hypothesis (1.19) on L convergence needed later on, Lemma 2.1 entails a rst set of yet quite basic a priori estimates.

whence (2.3), (2.4) and (2.5) result in view of (1.22) and (2.2).
To achieve further regularity information, especially on spatial gradients, besides the above we will make substantial use of a boundedness assumption on the ux coe cient functions bε which is yet weaker than the hypothesis (1.18) to be imposed in Theorem 1.2, but which already refers to essentially the same topology as the one addressed therein.

Turning weak into strong convergence. Proof of Theorem 1.3
Next approaching the core of our analysis, we intend to make sure that under the assumptions from Theorem 1.3, the very weak solutions obtained above are indeed weak solutions in the natural sense speci es as follows. which is such that ub ∈ L loc (Ω × [ , T); R n ) and f (·, ·, u) ∈ L loc (Ω × [ , T)), (3.2) and that

De
Here a crucial step will consist in passing to the limit ε ↘ in the respective second last summand in (3.3), which in view of (1.8) essentially amounts to turning the weak convergence feature in (1.24) into an appropriate statement on strong convergence. Our method of approaching this is in principle inspired by a strategy already pursued in previous studies (see e.g. [24,34,40]), namely intending to derive inequalities of the form by estimating the left-hand side therein directly through the weak inequality (1.12); in contrast to virtually all precedent cases, however, a major challenge in the present context stems from the circumstance that the integral inequality (1.12) merely addresses ln(u + ) rather than u itself, which seems to substantially impede appropriate testing procedures.
As a preparation for our main argument in this direction, to be detailed in the proof of Lemma 3.2, let us recall (cf. e.g. [40] for statements quite precisely covering the present situation) the well-known fact that for T > and ψ ∈ L p (Ω × (− , T ); R N ) with p ∈ [ , ∞] and N ∈ N, the Steklov averages S h ψ ∈ L p (Ω × ( , T ); R N ), h ∈ ( , ), as de ned by letting A(x, t) · ∇ ln(u + ) · ∇ ln(u + ) Proof. Without loss of generality we may assume that T be nite. For k ∈ N, we then let and using that ψ k : in Ω for all l ∈ N and ψ kl → ψ k a.e. in Ω as l → ∞, (3.8) and extend ψ k to a function ψ kl de ned on all of Ω × R by letting if x ∈ Ω and t ≥ T. (3.9) We furthermore abbreviate and so that actually L k is explicitly given by whence using that u ∈ L loc (Ω×[ , T)) and that ln( +ξ ) ≤ ξ for all ξ ≥ we conclude that besides the inclusion ln(u + ) · ψ k ∈ L loc (Ω × [ , T)) we also have L k (ψ k ) ∈ L loc (Ω × [ , T)), whereby it becomes possible to nd a null set N ⊂ ( , T) such that u(·, t ) ∈ L (Ω) for all t ∈ ( , T) \ N, (3.13) and that moreover each t ∈ ( , T) \ N is a common Lebesgue point of all the countably many mappings ( , T) t → Ω ln u(x, t) + ψ k (x)dx and ( , T) t → Ω L k (ψ k (x, t))dx for k ∈ N.
We nally observe that according to (3.7) and the representation (3.16), on the right of (3.23) we can simplify Similarly inserting (3.7) into (3.24) and (3.25), in view of the de nition (3.12) of L k we immediately conclude that (3.6) is a consequence of (3.23)-(3.28). Now if b complies with the regularity assumptions from Theorem 1.3, then the above can be combined with the convergence statements from Theorem 1.2 to deduce (3.4), and hence the desired strong approximation property, in the following sense.

Lemma 3.3. In addition to the assumptions from Theorem 1.2, suppose that
Then there exists a null set N * ⊂ ( , T) such that with u and (ε j k ) k∈N as given by Theorem 1.2 we have Proof. Since Lemma 2.5 especially entails that for a.e. t ∈ ( , T) we have uε(·, t ) → u(·, t ) in L (Ω) as ε = ε j k ↘ , (3.31) according to Lemma 3.2 we can pick a null set N * ⊂ ( , T) with the property that both (3.31) and (3.6) hold for each t ∈ ( , T) \ N * and all k ∈ N. Using that (3.31) in particular warrants that for any such t we know that u(·, t ) + belongs to L (Ω) and hence is nite a.e. in Ω, we see that whereas the validity of ≤ ln( + ξ ) ≤ ξ for all ξ ≥ (3.32) asserts the majorization ≤ u(·, t ) + Therefore, the dominated convergence theorem ensures that Ω u(·, t ) + + u(·,t )+ k · ln + u(·, t ) + k → as k → ∞ for all t ∈ ( , T) \ N * , (3.33) and quite a similar reasoning based on (1.22) shows that Next, once more relying on (3.31), by means of the l'Hospital rule we readily nd that in Ω as k → ∞, while thanks to (3.32), Again by the dominated convergence theorem, we thus obtain that (3.35) and that, similarly, Now on the right-hand side in (3.6), in order to adequately cope with the second summand we rst recall (1.15) and invoke Young's inequality to estimate |b| for all t ∈ ( , T).
using the Hölder inequality we see that here for all t ∈ ( , T), (3.38) and observe that the rst integrand on the right satis es u u+k q q− → a.e. in Ω × ( , t ) as k → ∞, and is Finally, two further arguments based on dominated convergence show that thanks to (1.9) and the inclusion f (·, ·, u) ∈ L α loc (Ω × [ , T)), as asserted by (1.24) in view of (1.7) and (1.8), t Ω g(x, t) and t Ω f (x, t, u) In summary, upon collecting (3.33)-(3.36) and (3.39)-(3.41) we obtain from Lemma 3.2 that where now making full use of (3.31) we see that due to (1.20) and (1.22), the right-hand side appears as a limit of the corresponding expressions associated with (1.14) in the sense that for all t ∈ ( , T) \ N * , Since moreover, again by dominated convergence, As furthermore t Ω f+(x, t, u) ≤ lim inf ε=ε j k ↘ t Ω f+(x, t, uε) for all t ∈ ( , T) due to (1.23) and Fatou's lemma, this means that actually which again in view of (1.23) implies that for any such t , f+(·, ·, uε) → f+(·, ·, u) in L (Ω×( , t )) as ε = ε j k ↘ .
Since (1.8) entails that in Ω for all ε ∈ (ε j ) j∈N , one nal application of a dominated convergence principle reveals that again by (1.23), Together with the weak convergence statement in (1.24), by uniform convexity of L α (Ω × ( , t )) for all t > this yields (3.30).

. Proof of Theorem 1.3
As a last preliminary for Theorem 1.3, let us state a chain rule type statement which should be essentially wellknown, but for which we include a brief argument as we could not nd a precise reference in the literature. Proof. For k ∈ N letting ρ k (ξ ) := min{e ξ , e k }, due to the Lipschitz continuity of ρ k we may invoke a wellknown version of the chain rule in W , (Ω) to infer from the inclusions e w ∈ L (Ω) and ∇w ∈ L (Ω; R n ) that ρ k (w) belongs to W , (Ω) with ∇ρ k (w) = χ {w<k} e w ∇w a.e. in Ω. (3.43) Accordingly, for integers k and l with l > k we can estimate whence again by hypothesis we conclude that (∇ρ k (w)) k∈N forms a Cauchy sequence in L (Ω; R n ). Since, on the other hand, clearly ρ k (w) → e w in L (Ω) as k → ∞ by Beppo Levi's theorem, we thus must have ∇ρ k (w) → ∇e w in L (Ω) as k → ∞, so that the claim results on observing that an application of the dominated convergence theorem to (3.43) directly shows that ∇ρ k (w) → e w ∇w in L (Ω) as k → ∞.
In order to make our general results derived above applicable to the present particular setting, for ε ∈ ( , ) we . Very weak solutions. Proof of Theorem 1.4 In light of the above observations, for an application of Theorem 1.2 it will thus be su cient to nd (ε j ) j∈N ⊂ ( , ) and v ∈ L loc ([ , ∞); W , (Ω)) such that ε j ↘ as j → ∞, and that ∇vε → ∇v in L loc (Ω × [ , ∞)) as ε = ε j ↘ . (4.4) This will be achieved through an analysis of the speci c systems (1.28) and (4.3), particularly focusing on the second equation therein as the main additional ingredient in comparison to (1.4) and (1.14), but in some places as well resorting to statements derived for the latter general setting in Section 2. A fundamental property of (4.3), for instance, has been achieved in Lemma 2.2 already: Lemma 4.1. Suppose that (1.29) holds with some k F > , K F > and α > . Then for all T > there exists Proof. Noting that our above selections warrant applicability of Lemma 2.2, we immediately obtain (4.5) from (2.4). By relying on appropriate smoothing properties of the inhomogeneous heat equation satis ed by vε, the previous lemma rstly entails a uniform spatial L bound for vε whenever α complies with the largeness assumption from Theorem 1.4.
The following statements on convergence of both uε and vε are thus rather evident. as ε = ε j ↘ .
Proof. In view of (4.11), Lemma 2.5 applies so as to yield the statements concerning uε along an appropriate sequence. Relying on the boundedness properties derived in Lemma 4.2) and Lemma 4.3, as well as on the Vitali convergence theorem, a straightforward further subsequence extraction based on the Aubin-Lions lemma thereafter enables us to achieve also (4.16) and (4.17) with some nonnegative v ful lling (4.15).
As an application of Theorem 1.2 will require strong, rather than merely weak, L convergence of bε = −∇vε, an additional consideration concerning this will be necessary: Proof. We x ε ∈ ( , ) and ε ∈ ( , ) and then obtain on taking di erences in the respective second equa- Here since ( u ε +ε u ε ) ε ∈(ε j ) j∈N is bounded in L α (Ω × ( , T)) by Lemma 4.1 and a.e. in Ω × ( , T) convergent to u according to (1.23), from Egorov's theorem it follows that uε Apart from that, from (4.16) we know that (4.20) once more because the hypothesis (4.6) warrants that α α− < n+ n . In view of (4.2) and (4.17) we hence infer by employing the Hölder inequality that for all ε ∈ ( , ), , (4.21) so that again relying on (4.2) and (4.20), and on the boundedness of ( uε +εuε ) ε∈( , ) in L α (Ω × ( , T)), as resulting from Lemma 4.1, we see that (4.18) is a consequence of (4.21). Thus having at hand all ingredients necessary for an application of Theorem 1.2, we can utilize the latter to obtain our main results on global very weak solvability in (1.28). P of Theorem 1.4. Taking (ε j ) j∈N , u and v as provided by Lemma 4.4, on the basis of the strong convergence result from Lemma 4.5 we may employ Theorem 1.2 to obtain a subsequence, again denoted by (ε j ) j∈N for notational convenience, along which for the solutions of (4.3) we have uε →ũ a.e. in Ω × ( , ∞) as ε = ε j ↘ , so that clearlyũ must coincide with u and hence u must have the claimed solution properties with regard to (1.4). In view of (1.12), the regularity features in (1.33) are therefore immediate by-products of Theorem 1.2 and Lemma 4.4, whereas the derivation of (1.34) can be chieved in a straightforward manner by taking ε = ε j ↘ in an accordingly tested version of the second sub-problem from (4.3).

. Weak solutions. Proof of Theorem 1.5
Next, in order to derive the stronger integrability property (1.27) required for an application of Theorem 1.3, beyond (4.1) and (4.2) we will assume that v ∈ D(A β ) and that sup ε∈( , ) A β v ε L (Ω) < ∞ (4.22) for some β ∈ ( , ), with A = −∆ + as introduced before the formulation of Theorem 1.5. Here we note that for any nonnegative v ∈ D(A β ), the requirements in (4.2) and (4.22) can simultaneously be ful lled with some (v ε ) ε∈( , ) ⊂ C (Ω; [ , ∞)) by e.g. xing m ∈ N such that m > n+ and letting v ε := ( + εA) −m v for ε ∈ ( , ), for instance: In fact, v ε then is nonnegative by order preservation of ( + εA) − , and the inclusion v ε ∈ C (Ω) is ensured by the fact that m > n+ warrants continuity of the embeddings D(A m ) → W m, (Ω) → C (Ω) ( [11]); apart from that, the L convergence property in  according to the Hölder inequality and the fact that α α− ≤ . Here we may rely on a standard interpolation result ( [9]) to infer from the inequalities β < β ≤ β+ , as ensured by the restrictions β > and β ≤ , that there exists c > ful lling for all t > .
When inserted into (4.25) and combined with Young's inequality, this shows that we can nd c > such that d dt for all t > , (4.26) where in the case β < we may two more times use Young's inequality to see that since As the resulting inequality evidently extends so as to remain valid also in the borderline case β = , from (4.26) we altogether obtain that Proof. We rst consider the case β = , in which due to elliptic regularity theory ( [9,10]) it is well-known that A β+ (·) L (Ω) = (−∆ + )(·) L (Ω) and A β (·) L (Ω) = ( ∇(·) L (Ω) + · L (Ω) de ne norms equivalent to · W , (Ω) and · W , (Ω) , respectively, so that by a Gagliardo-Nirenberg interpolation we nd c > and c > such that writing q := (n+ ) n we have for all ϕ ∈ D(A).
According to (4.27) and Young's inequality, we thus obtain that in this case, and that hence (4.28) results from Lemma 4.6.
If β < , however, we rst make use of the strivt inequality in (4.27) to x γ > β such that noting that then a known embedding result ( [11]) warrants that D(A γ ) → W ,q (Ω). As furthermore our assumption q > ensures that γ ≤ β + q ≤ β+ , once more according to an appropriate interpolation property of fractional powers ([9, Part 2, Theorem 14.1]) we can x c > and c > such that dt for all ε ∈ ( , ).
Observing that herein q(γ − β) ≤ thanks to the right inequality in (4.29), again invoking Lemma 4.6 we infer (4.28) from this. We can thereby proceed to make sure that our very weak solutions are in fact weak solutions whenever the hypotheses from Theorem 1.5 are met. If β ∈ ( n+ α , ), and hence α > n+ , observing that then we may pick any q ∈ [ α α− , (n+ ) n+ − β ) to similarly conclude on the basis of (1.37) that Theorem 1.3 and Theorem 1.4 imply the stated solution properties of (u, v). P of Corollary 1.6. We only need to apply Theorem 1.5 to α := , β := and q := , and once more make use of the well-known fact that then D(A β ) = D(A ) = W , (Ω).

Application to a Shigesada-Kawasaki-Teramoto system
We will next focus on the Shigesada-Kawasaki-Teramoto system (1.39) under the standing assumptions that d , d and µ are positive, that a , a , µ , a and a are nonnegative, and that u ∈ L (Ω) and v ∈ L ∞ (Ω) are nonnegative. (5.1) As approximations of (1.39) convenient for our purposes, for ε ∈ ( , ) we shall consider Finally, (5.9) can be derived from this in a standard manner by using (5.2) to see that for all ϕ ∈ C (Ω) with ∇ϕ L (Ω) + ϕ L (Ω) ≤ , Proof. The existence of a sequence (ε j ) j∈N and limit functions u and v with the properties in (5.11)-(5.14) immediately results from a straighforward extraction process based on Lemma 2.5, Lemma 5.2, (5.5) and the Aubin-Lions lemma. The veri cation of (1.41) can thereupon be achieved on testing the second equation in (5.2) by φ ∈ C ∞ (Ω × [ , ∞)) and observing that (5.12)-(5.14) are especially su cient for passing to the limit in each of the respective nonlinear contributions in the sense that On particularly choosing t = t k here, with (t k ) k∈N ⊂ ( , ∞) \ N ful lling t k ↗ ∞ as k → ∞, we easily infer from (5.19) that passing to a conveniently relabeled subsequence we can achieve that also ∇ρ(vε) → ∇ρ(v) a.e. in Ω × ( , ∞)) and thus, by for all t ∈ ( , ∞) \ N as ε = ε j ↘ , which together with (5.14) entails (5.15) when a > . In the case a = of linear di usion, the argument actually becomes much simpler and may thus be omitted here. We can thus apply Theorem 1.2 in a straightforward manner to achieve the claimed results on very weak solvability in (1.39). P of Theorem 1.7. Thanks to the strong convergence result from Lemma 5.4, in view of Theorem 1.2 and Lemma 5.3 we only need to make sure that gε := −µ a uε vε satis es gε → g in L loc (Ω × [ , T)) as ε = ε j ↘ for each T > . However, since gε → g a.e. in Ω × ( , ∞) by (5.11) and (5.12), and since |gε| ≤ c (T)uε in Ω × ( , T) with c (T) := µ a sup ε∈( , ) vε L ∞ (Ω×( ,T)) being nite due to (5.5), by means of the dominated convergence theorem this directly results from the L convergence property of (uε) ε∈(ε j ) j∈N in (5.10).
In conclusion, Theorem 1.3 can be applied so as to yield our claimed results on global existence of weak solutions in (1.39) for initial data merely belonging to L × (W , ∩ L ∞ ). P of Theorem 1.8. We let (ε j ) j∈N , u and v be as provided by Theorem 1.7. Then due to the fact that clearly T Ω |∇v| ≤ lim inf ε=ε j ↘ T Ω |∇vε| for all T > by Lemma 5.6, we may apply Theorem 1.3 to q := and α := and thereby infer the claimed additional regularity and solution properties, beyond those guaranteed by Theorem 1.7, of u. It thus remains to note that the inclusion v ∈ L ∞ loc ([ , ∞); W , (Ω)) is a by-product of (5.24) when once more combined with the fact that P from (5.23) satis es P ≥ d > throughout [ , ∞).