Can simultaneous density-determined enhancement of diffusion and cross-diffusion foster boundedness in Keller–Segel type systems involving signal-dependent motilities?

The reaction-(cross-)diffusion system }0,\hfill \\ {v}_{t}={\Delta}v-v+u,\quad \hfill & x\in {\Omega},\enspace t{ >}0,\hfill \end{cases}\end{equation}?> ut=Δ(umϕ(v)),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, is considered under no-flux boundary conditions in smoothly bounded convex domains Ω⊂Rn , where m ⩾ 1 and n ⩾ 2, and where ϕ generalizes the prototype obtained on letting }0,$?> ϕ(v)=a+b(v+d)−α,v>0, with a ⩾ 0, b > 0, d ⩾ 0 and α ⩾ 0. In this framework, it is firstly seen that if }\frac{n}{2},\alpha {< }\frac{nm-2}{n-2}\;\text{and}\;\alpha {< }\frac{2\left(m+1\right)}{n-2},$?> m>n2,α<nm−2n−2andα<2(m+1)n−2, then finite-time blow-up is excluded in the sense that for all suitably regular initial data an associated initial-boundary value problem admits a globally defined weak solution (u, v) with u being locally bounded in Ω¯×[0,∞) . Under the assumption that additionally α<(n+2)m−n+22(n−2), these solutions are moreover shown to be bounded throughout Ω × (0, ∞) in both components. In view of results known for the case m = 1, this particularly indicates that increasing m in () goes along with a certain regularizing effect despite the circumstance that thereby both the diffusion and the cross-diffusion mechanisms implicitly contained in () are simultaneously enhanced.


Introduction
We consider the parabolic system with m 1 and a given parameter function φ : (0, ∞) → (0, ∞). In recent literature on biomathematical modelling, systems of this form have been proposed as possible descriptions of the collective behaviour within bacterial populations in certain situations in which, according to experimental findings, the ability to perform random diffusive movement is significantly influenced by a signal substance secreted by the cells themselves, in essence particularly leading to reduced motility of individuals in regions of large signal concentrations ( [8,19]). Accordingly guided by the ambition to understand (1.1) in the presence of functions φ which depend on the chemical concentration v = v(x, t) in a nonincreasing manner, e.g. by generalizing simple functional laws of the form with a 0, b > 0, d 0 and α 0, the mathematical literature on (1.1) and close relatives has as yet concentrated on the case m = 1 in which with regard to the population density u = u(x, t) the considered diffusion process is essentially Brownian. In such settings and when posed along with homogeneous no-flux boundary conditions in n-dimensional bounded domains, (1.1) indeed has been found to admit global bounded classical solutions for widely arbitrary initial data when n = 2 and φ is suitably regular and uniformly positive, which in the context of (1.2) essentially reduces to the requirement that a be positive ( [28]). In three-and higherdimensional frameworks, such assumptions on non-degeneracy of diffusion at large values of v are merely known to warrant global existence of weak solutions, with a statement on smooth solvability and boundedness available only under appropriate restrictions on the size of the initial data ( [28]). In degenerate situations associated with the choice a = 0 in (1.2), results on global existence of classical solutions seem limited to the case when n = 1 and α > 0 is arbitrary ( [7]), whereas in multi-dimensional settings the literature is yet apparently restricted to weak solution frameworks, and to corresponding findings exclusively in the cases when either n = 2 and α < 2, or n = 3 and α < 4 3 ([7]). In this case m = 1, farther-reaching information especially on boundedness properties seems available only for variants of (1.1) either containing a simplified signal evolution mechanism, or additionally including the dissipative action of logistic-type growth restrictions. For instance, in a simplified variant of (1.1) in which the second equation therein is replaced with the elliptic equation 0 = Δv − v + u, global bounded classical solutions are known to exist for all reasonably regular initial data whenever n 1 and φ satisfies (1.2) with a = 0, b > 0, d = 0 and α < 2 (n−2) + ( [1]). If logistic terms of the form ρu − μu 2 with ρ > 0 and μ > 0 are added to the first equation in the fully parabolic model (1.1) with m = 1, actually any choice of α > 0 in (1.2) leads to global existence of bounded smooth solutions when n = 3 ( [15], cf also [30]), and even some statements on stabilization toward homogeneous equilibria available if μ is suitably large ( [15,20]; cf also [21][22][23] for similar results on related systems involving superquadratic degradation terms).
The case m > 1: enhancement of diffusion and cross-diffusion in a Keller-Segel system. The purpose of the present study is to provide a first step toward an understanding of how far an increase of the parameter m may regularize (1.1) in the sense of blow-up suppression, especially in situations in which for the case m = 1 either only weak and possibly quite irregular solutions are known to exist, or any result on global solvability is lacking at all.
To put this question into a slightly broader perspective, let us observe that (1.1) can be rewritten according to and that hence (1.1) formally corresponds to a Keller-Segel type cross-diffusion system with diffusion and cross-diffusion rates which, besides depending on u whenever m > 1, are both explicitly influenced by v. Now a rich literature indicates that with regard to the occurrence of singularity phenomena, at least in some subclasses of (1.3) with more general ingredients the behaviour of S(u, v) relative to D(u, v) especially at large values of u plays a decisive role. Specifically, when D = D(u) and S = S(u) are independent of the signal concentration, the corresponding Neumann problem for (1.3) is known to admit globally bounded smooth solutions for all suitably regular initial data if D and S, besides complying with some technical assumptions which in substance mainly reduce to the mere requirement that D(u) neither grows nor decays faster than algebraically as u → ∞, satisfy the subcriticality inequality lim sup A certain optimality of this result ( [13,14,27]), which has partially been extended to cases of exponentially decaying D ( [6,35]), is indicated by several findings on the occurrence of some unbounded radial solutions in balls under assumptions on D and S which essentially complement that in (1.4) by supposing that, in different concrete flavours available in the literature, D(u) grows somewhat faster than u 2 n as u → ∞ ( [5,32,34,36]). In contrast to this quite comprehensive picture, only little seems known with respect to the obvious question to which extent features of the latter dichotomy-like flavour persist also in the presence of signal dependencies in the migration rates entering (1.3), with the few examples available in the literature mainly concentrating on the derivation of global existence and boundedness results in the application-relevant special case when D ≡ 1 and S(u, v) = χ u v with χ > 0 (cf e.g. [3,10,18,26,33] and also [16] for a slightly more distant relative). In particular, the literature seems to have left widely unclarified how for the ratio D S might retain relevance in this regard also when D and S depend on v. Main results. When viewed against this background, the outcome of this work may be summarized as indicating that in the special setup defined through (1.4), with regard to boundedness properties of solutions the system (1.3) might exhibit features quite different from those known for the signal-independent case in which (D, S) = (D, S)(u). Namely, we shall see that despite the fact that then S(u,v) D (u,v) 2 , and that thus the findings around (1.5) suggest linear and hence explosion-supercritical growth of S D at large values of u, in the presence of suitably large m > 1 the particular system structure in (1.1), and especially the precise link between D and S in (1.3) via (1.4), allows for the construction of global solutions with favourable boundedness properties for arbitrarily large initial data, and under quite mild assumptions on φ which inter alia include degenerate cases. Specifically, we shall consider in a bounded domain Ω ⊂ R n with smooth boundary, where n 2 and m > 1, and where φ generalizes the prototypical choice from (1.2) in that with a certain number α 0. The first of our main results then asserts global existence of a solution locally bounded in Ω × [0, ∞), provided that m is suitably large and α is appropriately small: and that φ satisfies (1.7)-(1.9) with some α 0 satisfying (1.12) Then for any pair (u 0 , v 0 ) of initial data u 0 ∈ W 1,∞ (Ω) and v 0 ∈ W 1,∞ (Ω) fulfilling u 0 0, u 0 ≡ 0 and v 0 > 0 in Ω, the problem (1.6) admits at least one global weak solution (u, v), in the sense of definition 2.1 below, which has the additional properties that v ∈ C 0 (Ω × [0, ∞)), and that for each T > 0 one can find C(T) > 0 such that Under a slightly stronger assumption on the parameter α, each of these solutions is bounded actually throughout Ω × (0, ∞).
Then there exists C > 0 such that the global weak solution of (1.6) from theorem 1.1 additionally satisfies Remark.
(a) We emphasize that in the particular case n = 2, any choice of m > 1 and α 0 is admissible both in theorems 1.1 and 1.2, hence implying that in this planar situation, any superlinear porous medium type enhancement of diffusion and cross-diffusion considerably increases the knowledge about global boundedness in comparison to the findings achieved in [7] only for α < 2 and only in contexts of possibly unbounded weak solutions. (b) If α satisfies both (1.11) and (1.12), then necessarily α < 1 2 · ( nm−2 n−2 + 2(m+1) n−2 ) = (n+2)m 2(n−2) , so that the hypotheses from theorem 1.2 are indeed stronger than those underlying theorem 1.1 whenever n 3. (c) In view of the diffusion degeneracy near points where u = 0 whenever m > 1, it seems that classical solutions to (1.6) can in general not be expected, and that hence resorting to appropriately generalized frameworks of solvability, such as done in theorem 1.1, indeed appears in order.

Main ideas.
In order to suitably capture the structural information encoded in (1.6) and the particular liaison between diffusion and cross-diffusion therein, in a first step we shall pursue the goal of deriving some fundamental regularity information through a duality-based argument in the style of reasonings which are quite well-established in the context of reactiondiffusion systems free of cross-diffusion ( [4,17]), but quite a simple form of which has also been underlying the analysis performed in [28] for the case m = 1. For general m 1 and φ satisfying (1.7) and (1.9) with some α 0, this will firstly lead to an inequality of the form with a certain C = C(m, φ) > 0, where A denotes the self-adjoint realization of −Δ + 1 under homogeneous Neumann boundary conditions in L 2 (Ω), and where (u ε , v ε ) denotes the global solution of a suitably regularized variant of (1.6) for ε ∈ (0, 1) (lemma 3.1). By means of suitable interpolation arguments, for T > 0 this will be seen to imply estimates of the form for allt ∈ (0, T) and ε ∈ (0, 1), (1.19) provided that α 0 satisfies (1.12) and the condition α < nm n−2 slightly stronger than (1.11), with K(T) actually independent of T if additionally (1.15) holds (lemma 3.5).
Next relying on the hypotheses on α from theorems 1.1 and 1.2 in their full strength, we shall thereafter see that under the additional assumption m > n 2 it is possible to apply an iterative L p estimation procedure to the respective second equation in order to turn (1.19) into bounds for Ω v p ε with arbitrary p > 1 (corollary 4.4). Using that (1.19) thus actually implies corresponding boundedness properties of u ε in space-time L p norms with arbitrary p ∈ (1, m + 1) (corollary 4.5), we can thereafter derive estimates for v ε in W 1,q (Ω) with some q > n (lemma 4.6), which will finally be seen to entail L ∞ bounds for u ε (lemma 5.3). Both theorems 1.1 and 1.2 can hence be verified on the basis of straightforward extraction procedures.

Preliminaries. Global approximate solutions
In order to substantiate our goal, let us begin by specifying the concept of solvability to be pursued in the sequel.
and if as well as (2.4) Our path toward the construction of a solution in this framework will be based on a regularization of (1.6) which for convenience we plan to design in such a way that approaches well-known in the theory of quasilinear Keller-Segel type systems become applicable so as to assert global smooth solvability thereof. To this end, let us fix a number and, for ε ∈ (0, 1), consider the problem in which apparently the degeneracy of cell diffusion at vanishing population densities, as present in (1.6) whenever m > 1, is removed. Apart from that, due to (2.5) and strength of diffusion enhancement thereby induced, in each of these problems also finite-time blow-up phenomena can be ruled out, as seen in lemma 2.4 below on the basis of the following essentially well-known global solvability feature of quite general quasilinear chemotaxis systems with subcritical sensitivities.
with some k D > 0, K D > 0, γ ∈ R and Γ ∈ R, and that there exist C > 0 and η > 0 such that Then for any nonnegative (2.8) and such that both u and v are nonnegative in Ω × (0, ∞).
Proof. Based on well-established theories of local existence and extensibility, as contained in [2] for rather general second-order cross-diffusion systems and e.g. in [36] for a more specific setting close to that in (2.7), this can be derived by performing evident minor adaptations to the a priori estimation procedure applied to merely u-dependent functions D and S in [27], so that we may refrain from giving details here. In order to make (2.6) accessible to the latter especially in cases when φ is irregular near v = 0, let us reformulate a favourable a priori positivity property of externally forced linear heat equations even in the presence of linear degradation, as implicitly contained in [37, lemma 2.2] already but concretized there in a slightly different setting. This is the only place where convexity of Ω is explicitly referred to in this paper.

(2.9)
Proof. According to the convexity of Ω, we can find c 1 > 0 such that for the Neumann heat semigroup (e tΔ ) t 0 on Ω we have e tΔ ϕ c 1 Ω ϕ for all t > 1 and each nonnegative ϕ ∈ C 0 (Ω) ( [9,12]). By nonnegativity of h and the comparison principle, in the variation-of-constants representation As furthermore, again by the comparison principle, from (2.10) we infer (2.9) upon an evident choice of Λ(Ω). By suitably combining lemma 2.3 with lemma 2.2 we can now make sure that indeed (2.6) is globally solvable for each ε ∈ (0, 1). (2.11) and there exists C > 0 fulfilling v ε (x, t) C for all x ∈ Ω and any t > 0. (2.12) Proof. With Λ(Ω) > 0 taken from lemma 2.3, we let and using that c 1 is positive thanks to our hypotheses on u 0 and v 0 we may fix a positive function , observing that then (1.8) ensures that both φ and φ are bounded on [0, ∞ ). Then for fixed ε ∈ (0, 1), define functions D and S which belong to C 2 ([0, ∞) 2 ) with S(0, v) = 0 for all v 0, and since m 1 and M m by (2.5), we can estimate whereas clearly As the latter moreover implies that using that (2.5) guarantees that m − M + 1 < 2 n we may invoke lemma 2.2 to infer the existence of a global classical solution (u, v) = (u ε , v ε ) to the accordingly obtained problem (2.7), enjoying the regularity features in (2.8), and nonnegative in both its components.
An integration of the first equation in (2.7) thereafter shows that (2.11) holds, which in turn, through lemma 2.3, entails that thanks to our definition of c 1 , and that hence our definitions of D and S warrant that indeed (u ε , v ε ) solves (2.6) and satisfies (2.12) with C := c 1 .
Without further explicit mentioning, throughout the sequel we shall assume that (1.7) and (1.8) be satisfied, that u 0 and v 0 satisfy the requirements from theorem 1.1, and that M is such that (2.5) holds, and let ((u ε , v ε )) ε∈(0,1) denote the family of approximate solutions obtained in lemma 2.4.
Forming a last preliminary, let us formulate a rather straightforward consequence of (2.11) for a first ε-independent regularity feature of the respective second solution components.
Proof. Relying on (2.11), this can be seen by straightforward application of well-known smoothing properties of the Neumann heat semigroup (e tΔ ) t 0 on Ω: in fact, without loss of generality assuming that p > 1 we may invoke standard regularization estimates therefore ([31, p ) dσ is finite due to the fact that n 2 (1 − 1 p ) < 1 thanks to the assumption p < n n−2 , this already establishes (2.13).

Space-time L 1 estimates for u m+1 Ú −« via a duality argument
Our next goal will consist in making appropriate use of the particular structure of the first equation in (2.6) and the particular link between the diffusion and cross-diffusion mechanisms contained therein, and our strategy in this regard will follow classical duality-based arguments (cf e.g. [4,17] for related reasonings in more general frameworks, and [28] for a precedent addressing a less degenerate variant of (1.6)). To prepare an appropriate setup for our analysis in this direction, we let A denote the realization of −Δ + 1 under homogeneous Neumann boundary conditions in L 2 (Ω), with its domain thus given by D(A) = {ψ ∈ W 2,2 (Ω) | ∂ψ ∂ν | ∂Ω = 0}, and recall that A is self-adjoint and possesses a family (A β ) β∈R of corresponding densely defined self-adjoint fractional powers. Now a rather straightforward pursuit of duality-guided ideas leads to a first observation concerning the time evolution of A − 1 2 (u ε + 1) which in its dissipated part contains a functional that with respect to u ε grows in a considerably superlinear manner. Lemma 3.1. Assume that n 2 and m 1, and that (1.9) holds with some α 0. Then there exists C > 0 such that , from a corresponding embedding inequality and standard elliptic regularity in L p (Ω) ( [11]) we obtain c 1 > 0 and c 2 > 0 such that for all ψ ∈ W 2,p (Ω) such that ∂ψ ∂ν and thereafter we twice employ Young's inequality to infer the existence of c 3 > 0 such that and that since M + 1 > p > 1 we may pick c 4 > 0 such that Next, as a consequence of (2.12), we can find c 5 > 0 fulfilling v ε c 5 in Ω × (0, ∞) for all ε ∈ (0, 1), and note that due to (1.8) this firstly implies that with some c 6 > 0 we have φ(v ε ) c 6 in Ω × (0, ∞) for all ε ∈ (0, 1), (3.5) and that thanks to (1.9) and (1.7) this secondly entails the existence of c 7 > 0 satisfying As a final preparation, we once more draw on Young's inequality to fix c 8 > 0 such that and now make use of all these selections as follows: since ∂ t (u ε + 1) = u εt , from (2.6) we obtain the identity which we test by u ε + 1 to see by self-adjointness of A − 1 2 and of A −1 , for all t > 0 and ε ∈ (0, 1). Here due to (3.3), (3.2), (3.4) and (2.11), ε for all t > 0 and ε ∈ (0, 1) (3.9) with c 9 := c 4 u 0 + 1 M+1 L 1 (Ω) . Furthermore, combining (3.7) with (3.5) we see that for all t > 0 and ε ∈ (0, 1), whence (3.8) and (3.9) entail that 9 ε for all t > 0 and ε ∈ (0, 1).
Since finally estimating u ε + 1 max{u ε + ε, ε} and using (3.6) shows that for all t > 0 and ε ∈ (0, 1), this establishes (3.1) if we let C := max{ 1 c 7 , 2c 8 , 2c 9 }, for instance. In order to appropriately estimate the integral on the right of (3.1) in terms of the second summand on the left-hand side therein, to be accomplished in lemma 3.3, but to furthermore prepare an argument revealing dominance of the latter over the expression Ω |A − 1 2 (u ε + 1)| 2 (see lemma 3.4), let us state the following consequence of a simple interpolation based on the Hölder inequality. one can find C(p) > 0 such that p m+1 + C(p) for all t > 0 and ε ∈ (0, 1).

(3.11)
Proof. We first note that due to (3.10), we particularly have p < m + 1, so that setting q := pα m − p + 1 defines a nonnegative number which, by making full use of (3.10) now, can be seen to satisfy and hence q < n n−2 . Therefore, lemma 2.5 applies so as to yield c 1 > 0 such that Ω v q ε c 1 for all t > 0 and ε ∈ (0, 1), which we combine with the Hölder inequality, applicable since, still, p < m + 1, to estimate for all t > 0 and ε ∈ (0, 1).
In fact, under assumptions on α actually less restrictive than those in theorem 1.1 the latter can be combined with adequate embedding properties to suitably bound the right-hand side in (3.1). then for all η > 0 there exists C(η) > 0 such that for all t > 0 and ε ∈ (0, 1).
A second application of lemma 3.2 will eventually enable us to turn (3.1) into an autonomous ODI containing a linear absorption, provided that the stronger hypotheses on α from theorem 1.2 are met. Lemma 3.4. Assume that n 2 and m 1, and that (1.9) is satisfied with some α 0 such that (1.15) holds. Then there exists C > 0 such that ε (·, t)v −α ε (·, t) + C for all t > 0 and ε ∈ (0, 1).

Estimating v ε in L p (Ω) for arbitrary finite p
We next intend to turn the weighted L m+1 estimate for u ε provided by lemma 3.5 into bounds for v ε suitably improving those from lemma 2.5. In view of the particular structure of the integrand in (3.24), and especially its dependence on v ε , for an efficient exploitation thereof it seems promising to act, at a first stage, in the context of standard L p testing procedures. The information thereby gained, consisting in L p bounds for v ε in arbitrary L p spaces not only in the case n = 2 (see lemma 2.5) but also when n 3, will finally enable us to essentially neglect the factor v −α ε in (3.24), and to use the resulting version thereof in the derivation of gradient bounds for v ε through more straightforward semigroup estimates.
The following outcome of [25, lemma 3.4] will be needed in our first step in this direction, to be established in lemma 4.2. Then for all t ∈ (0, T).
By means of the latter and an appropriate testing procedure, we can derive the following core of an iterative step potentially improving our regularity information on v ε . Here our assumption (4.2) on u ε is formulated in such a way that both alternatives possible in (3.25) can conveniently be included without explicit reference to requirements on α which are actually not needed in this part. Then given any p > p 0 fulfilling for all t ∈ (0, T) and ε ∈ (0, 1), (4.5) and that Proof. We abbreviate t) for t > 0 and ε ∈ (0, 1), and given p > p satisfying (4.3) and (4.4) we first test the second equation in (2.6) by v p−1 ε to see that due to Young's inequality, for t > 0 and ε ∈ (0, 1). (4.7) Here in the case when incidentally (m+1)(p−1)+α m p , again by means of Young's inequality we can utilize (4.1) to obtain that for each fixed T > 0, and ε ∈ (0, 1).
In order to recursively show that for each j 0 and p ∈ [p j , p j+1 ) one can find K (p) : (0, ∞) → (0, ∞) such that (4.16) holds and that (4.15) is valid for all T > 0, (4.18) we first recall lemma 2.5 to see that for each p ∈ [0, n n−2 ) we can find c 1 (p) > 0 fulfilling If the property in (4.18) has already been asserted for any integer up to j − 1 with some j 1, however, then we note that by (4.17) and the inequalities n − 2m − 2 < 0 and p j+1 > n n−2 we have because α < nm n−2 by (1.11). By means of an argument based on continuous dependence, we can therefore pick p j+1 ∈ (p j , p j+1 ) such that whence in view of the assumed validity of the statement in (4.18) for j − 1, lemma 4.2 applies so as to show that for any such p we can find K (p) : (0, ∞) → (0, ∞) satisfying (4.15) for all T > 0, as well as (4.16). Since Ω is bounded, this already implies the property claimed in (4.18) throughout the entire interval [p j , p j+1 ), and hence completes the verification of (4.18).
It remains to observe that ∪ j 0 [p j , p j+1 ) = [1, ∞) to infer that the infinite collection of statements contained in (4.18) entails the intended conclusion.
In light of lemmas 2.5 and 3.5, from the latter we obtain the intended main result concerning integral estimates for v ε . Proof. If n 3, this readily results upon combining lemma 4.3 with lemma 3.5, whereas in the case n = 2 we only need to recall lemma 2.5.

An estimate including ∇v ε
Having corollary 4.4 at hand, from lemma 3.5 we can immediately draw the following conclusion concerning space-time integrability properties of u ε without the appearance of weight functions.

L ½ estimates for u
Strongly relying on the possibility to choose the exponent in lemma 4.6 to satisfy q > n, we can next achieve estimates for u ε with respect to the norms in L p (Ω), firstly for arbitrary finite p (lemma 5.2) and then for p = ∞ (lemma 5.3), on the basis of a rather straightforward and again variational-type procedure. This will be launched by the standard computation underlying the following basic step toward this, which will independently be used also in our deduction of regularity features enjoyed by spatial and temporal derivatives of u ε (cf lemmas 6.1 and 6.2). Lemma 5.1. Let p > 0 and ψ ∈ C ∞ (Ω). Then ∇ψ for all t > 0 and ε ∈ (0, 1).
Proof. This can be verified by straightforward computation based on several integrations by parts in the first equation from (2.6). Due to lemma 4.6, for arbitrary p > 1 the identity obtained from (5.1) on taking ψ ≡ 1 can be turned into the following information on L p regularity of (u ε ) ε∈(0,1) .

First-order regularity properties of u
In view of the nonlinear nature of the diffusion process in the first equation from (1.6), and the accordingly nonlinear manner in which u appears in (2.3), it seems in order to supplement the bounds for u ε provided through lemma 5.3 by some further ε-independent regularity information capable of implying at least some pointwise convergence properties of (u ε ) ε∈(0,1) along subsequences. This will be the objective of the next two lemmata which, again on the basis of lemma 5.1, prepare an argument based on an application of an Aubin-Lions type lemma to (u ε + ε) β with suitably chosen β > 0 in lemma 7.1 below. We first concentrate on the spatial gradient, for which we obtain the following. Proof. We fix any β > m−1 2 and k ∈ N such that k > n 2 , and then infer from lemma 6.1, (2.11) and lemma 6.2 that for all T > 0, (u ε + ε) β ε∈(0, 1) is bounded in L 2 ((0, T); W 1,2 (Ω)), and that ∂ t (u ε + ε) β ε∈(0, 1) is bounded in L 1 (0, T); (W k,2 (Ω)) .
To finally obtain our main results, we only need to summarize: