Suppressing blow-up by gradient-dependent flux limitation in a planar Keller–Segel–Navier–Stokes system

The flux-limited Keller–Segel–Navier–Stokes system nt+u·∇n=Δn-∇·(nf(|∇c|2)∇c),ct+u·∇c=Δc-c+n,ut+(u·∇)u=Δu+∇P+n∇Φ,∇·u=0,(⋆)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lcl} n_t + u\cdot \nabla n &{}=&{} \Delta n - \nabla \cdot \Big ( n f(|\nabla c|^2) \nabla c\Big ), \\ c_t + u\cdot \nabla c &{}=&{} \Delta c - c + n, \\ u_t + (u\cdot \nabla ) u &{}=&{} \Delta u + \nabla P + n\nabla \Phi , \qquad \nabla \cdot u=0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$\end{document}is considered in a smoothly bounded domain Ω⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^2$$\end{document}. It is shown that whenever the suitably smooth function f models any asymptotically algebraic-type saturation of cross-diffusive fluxes in the sense that |f(ξ)|≤Kf·(ξ+1)-α2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |f(\xi )| \le K_f\cdot (\xi +1)^{-\frac{\alpha }{2}} \end{aligned}$$\end{document}holds for all ξ≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi \ge 0$$\end{document} with some Kf>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_f>0$$\end{document} and 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}, for any all reasonably regular initial data a corresponding no-flux/no-flux/Dirichlet problem admits a globally defined classical solution which is bounded, inter alia, in L∞(Ω×(0,∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Omega \times (0,\infty ))$$\end{document} with respect to all its components. By extending a corresponding result known for a fluid-free counterpart 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}), this confirms that with regard to the possible emergence of blow-up phenomena, the choice f≡const.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\equiv const.$$\end{document} retains some criticality also in the presence of fluid interaction.


Introduction
The choice f ≡ K f = const. in the two-dimensional version of the chemotaxis model n t = Δn − ∇ · nf (|∇c| 2 )∇c , τ c t = Δc − c + n, (1.1) with some K f > 0 and α > 0, namely, then all reasonably regular initial data of arbitrary size lead to global bounded solutions both when τ = 0 and when τ = 1 ( [28,45,46]). A natural next task now seems to consist in determining how far such criticality features persist when single two-component systems of the form (1.1) are embedded into more complex models. Here in accordance with considerable developments in both the modeling and the analysis-focused literature, of particular interest appear couplings of chemotactically migrating populations to liquid environments ( [8,9,24,33,34]), and indeed some significant effects of fluid interaction have either been indicated by numerical evidence ( [23,34]), or even been verified rigorously ( [10,16,21,22]). In order to address this topic for possibly flux-limited Keller-Segel systems of the form in a framework consistent with the modeling approach in [34], concentrating on the prototypical situation of bacterial populations coupled to a surrounding liquid through transport and buocancy we shall subsequently consider the problem ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n t + u · ∇n = Δn − ∇ · nf (|∇c| 2 )∇c , x ∈ Ω, t > 0, c t + u · ∇c = Δc − c + n, x ∈ Ω, t > 0, u t + (u · ∇)u = Δu + ∇P + n∇Φ, ∇ · u = 0, x∈ Ω, t > 0, ∂n ∂ν = ∂c ∂ν = 0, u = 0, x ∈ ∂Ω, t > 0, n(x, 0) = n 0 (x), c(x, 0) = c 0 (x), u(x, 0) = u 0 (x), x∈ Ω, (1.3) in a smoothly bounded domain Ω ⊂ R 2 , with a suitably regular chemotactic sensitivity coefficient f and a given gravitational potential Φ. Within this setting and that of a corresponding parabolic-elliptic simplification, the first of the above two criticality features of f ≡ const. has already been investigated numerically and analytically: While simulations predict that fluid coupling of this type may entail some blow-up delay by enforcing boundedness in the presence of large-mass initial data for which solutions to (1.1) blow up ( [23]), a result on global solvability for small-mass initial data indicates that the corresponding critical mass phenomenon for (1.1), albeit with a possibly changed particular threshold value, persists also under the influence of the considered fluid interplay ( [43]).

Main results
The purpose of the present study now consists in examining corresponding stability properties of the second among the mentioned aspects of criticality related to the choice of constant f in (1.1). Specifically concerned with the question whether in the fully coupled model (1.3) still an arbitrarily small algebraic flux limitation of the form in (1.2) is sufficient to suppress any unboundedness phenomenon, by means of an analytical approach quite completely different from those pursued in the literature on (1.1) we shall find the following essentially affirmative answer. In its formulation, as well as throughout the sequel, we let W 1,2 0,σ (Ω) := W 1,2 0 (Ω; R 2 ) ∩ L 2 σ (Ω) with L 2 σ (Ω) := {ϕ ∈ L 2 (Ω; R 2 ) | ∇ · ϕ = 0}, and letting P represent the Helmholtz projection on r>1 L r (Ω; R 2 ) we let A = −PΔ and (A ϑ ) ϑ>0 denote the Stokes operator in L 2 (Ω; R 2 ), with domain given by D(A) := W 2,2 (Ω; R 2 ) ∩ W 1,2 0,σ (Ω), and the family of its corresponding fractional powers, respectively. Theorem 1.1. Suppose that Ω ⊂ R 2 is a bounded convex domain with smooth boundary, and assume that Φ ∈ W 2,∞ (Ω), and that f ∈ C 2 ([0, ∞)) is such that f ∈ C 2 ([0, ∞)) satisfies (1.2) with some K f > 0 and some α > 0. Moreover, suppose that (

1.4)
Then there exist uniquely determined functions which are such that n > 0 and c ≥ 0 in Ω × (0, ∞), and that with some P ∈ C 1,0 (Ω × (0, ∞)), the quadruple (n, c, u, P ) is a classical solution of (1.3) in Ω × (0, ∞). Moreover, this solution is bounded in the sense that one can find q > 1 and ϑ ∈ ( 1 2 , 1) satisfying By thus essentially asserting persistence of a boundedness feature, as known to be enjoyed by a chemotaxis system, under the additional inclusion of fluid interaction, this result appears to be quite in line with some precedent studies concerned with variants of (1.1) involving migration rates which depend on population densities, rather than signal gradients. In fact, the substantially more favorable mathematical structure of such systems has allowed for quite a complete knowledge concerning blow-up dichotomies, and for a widely comprehensive transfer of results from corresponding taxis-only systems to associated Keller-Segel fluid variants, both in cases governed by linear cell diffusion ( [35][36][37]41]) and in models involving certain nonlinear diffusion mechanisms ( [5][6][7]30,32,38,44]).

Main ideas
In contrast to the simple situation of (1.1), any derivation of boundedness features for solutions to (1.3) evidently needs to adequately cope with the circumstance that regularity properties of the crucial quantity ∇c may potentially be affected by the fluid field which is a priori unknown. Accordingly, this seems to limit accessibility to well-established direct strategies toward establishing L p bounds for n, e.g. by controlling the time evolution in expressions of the form as having played key roles in studies on chemotaxis systems for various choices of a > 0 and q ≥ 1 ( [20,32,47]). In fact, at a first stage the only source for regularity information on u appears to be the standard energy identity for the Navier-Stokes subsystem of (1.3), any meaningful application of which, however, seems to require integrability features of the forcing term n∇Φ therein that go beyond those trivially implied by mass conservation in the first solution component. A first and crucial step in our analysis will accordingly consist in the derivation of some additional basic estimates for n which do not rely on any features of the fluid field beyond its mere solenoidality, and which provide integrability properties sufficient to allow for a successful initial regularity analysis of u (Lemma 3.3). Fortunately, in Sect. 3 we shall see that precisely when α > 0, the assumption (1.2) warrants that this can indeed be achieved by tracing the evolution of for some p ∈ (0, 1), arbitrarily large r and suitably chosen M > 0 (cf. Lemma 3.6); although in its principal design related to the strategy from [35] where similar sublinear powers have been essential, this procedure deviates from the latter by simultaneously providing bounds for c which will later on come in handy when an apparently novel interpolation inequality will be applied (Lemmas 4.1, 4.5 and 4.6). Only after this preparation, Sect. 4 will reveal that an analysis of higher regularity features can be based on the examination of functionals in the style of those from (1.7) with arbitrarily large p > 1 and q > 1, when augmented by an added Dirichlet integral of the fluid velocity. In fact, appropriately estimating, respectively, appearing ill-signed contributions will enable us to derive temporally uniform bounds for the coupled functionals under consideration (Lemma 4.8), and to thereupon obtain our main result by means of a final bootstrap argument to be performed in Sect. 5.

Local existence and extensibility
Let us first note that according to approaches meanwhile quite standard, unique local-in-time classical solutions enjoying a fairly handy entensibility feature always exist:

1)
and that (n, c, u, P ) forms a classical solution of (1.3) in Ω × (0, T max ) with some appropriate P ∈ C 1,0 (Ω × (0, T max )). This solution has the additional property that Proof. This can be seen by straightforward adaptation of well-known arguments from the literature on local existence, extensibility, positivity and mass conservation in related chemotaxis-fluid systems, as detailed in [39], for instance.

Basic bounds for n and c: analyzing the Navier-Stokes energy
The first objective of this section is to make sure that functionals of the form in (1.8) enjoy certain entropy-like features when the integrability exponent p ∈ (0, 1) appearing therein is suitably small. Besides implying bounds for c in L r spaces with arbitrarily large r, this will reveal some additional regularity features of n that will turn out to be sufficient for a successful analysis of the standard energy identity associated with the Navier-Stokes subsystem of (1.3). Our analysis in this direction is launched by the following observation which already relies in an essential manner on the flux limitation mechanism expressed in our overall assumption (1.2).

ZAMP
Keller-Segel-Navier-Stokes system Page 5 of 24 72 Lemma 3.1. Assume (1.2) with some K f > 0 and α > 0, and let p ∈ (0, 1) be such that Then there exists C = C(p) > 0 such that Proof. We make use of the positivity of n in Ω × (0, T max ) to see upon integrating by parts in the first equation from (1.3) and using Young's inequality along with (1.2) that Here if α ≥ 1, then by the Hölder inequality and (2.2), abbreviating C 1 := Ω n 0 we have while if α ∈ (0, 1) and p ≤ α, then after employing Young's inequality we may proceed similarly to estimate for all t ∈ (0, T max ), so that in both these cases, (3.2) is a consequence of (3.3). If α < 1 and α < p ≤ α 1−α , finally, then in view of the Gagliardo-Nirenberg inequality we can pick for all ϕ ∈ W 1,2 (Ω), we therefore obtain that for all t ∈ (0, T max ), so that since, clearly, due to the fact that p(1−α) α ≤ 1, we readily infer (3.2) from (3.3) also for such p.
Fortunately, the ill-signed contribution to the right-hand side of (3.2) is dominated by the dissipation rate appearing in a zero-order testing procedure which operates on the second equation in (1.3), but which unlike that to be pursued later when analyzing functionals of the form in (1.7) (see Lemma 4.3) does not presuppose any quantitative regularity feature of the fluid field.
Proof. We fix q = q(p, r) > 1 such that

5)
ZAMP Keller-Segel-Navier-Stokes system Page 7 of 24 72 and observe that then the inequalities r > p + 1 > 1 ensure that q < 1 1−p and q < r, and that hence pq As furthermore clearly 2q p > 2, we may interpolate using the Gagliardo-Nirenberg inequality to find and Moreover, relying on the left inequality in (3.6) we may invoke Young's inequality to see that if we abbreviate C 3 := Ω n 0 , then with some for all a ≥ 0 and b ≥ 0, (3.9) and to appropriately make use of these selections, we multiply the second equation in (1.3) by (c + M ) r−1 and integrate by parts to find that Here by Young's inequality, while combining the Hölder inequality with (3.7) and (3.9), we infer that according to (2.2) and our definition of C 3 , We now employ (3.8) and then use that according to (3.5 and hence θ > 1, to see once more by means of Young's inequality that Together with (3.12) and (3.11) inserted into (3.10), this shows that and thereby establishes (3.4), because r − (r − 2) − 1 = 1, and because thanks to our restrictions r ≥ 3 and M ≥ 1, Among the consequences implied by an adequate linear combination of the inequalities provided by the latter two lemmata, the following will be referred to below.
Proof. We first apply Lemma 3.1 to fix and then rely on Lemma 3.2 to find Therefore, and so that by nonnegativity of h, As −y(t) ≤ Ω n p (·, t) and due to the restriction τ ≤ 1. This implies (3.14), whereas (3.13) results from (3.16) together with (3.17).
Through an interpolation along with (2.2), the weighted first-order estimate in (3.14) provides a bound for n with respect to some spatio-temporal Lebesgue norm that combines square integrability in time with some superlinear summability feature in space: ds ≤ C for all t ∈ (0, T max − τ ), (3.18) where again τ := min{1, 1 2 T max }.
Proof. Using that 4 p(2−p) > 2 p , we employ the Gagliardo-Nirenberg inequality to fix C 1 = C 1 (p) > 0 such that which when applied to ϕ := n p 2 shows that t+τ t n(·, s) 2 according to (2.2). The claim therefore results from the estimate in (3.14).
In order to make appropriate use of the latter in the context of the announced analysis of the Navier for all t ∈ (0, T ).
By means of the latter, we can now utilize the outcome of Corollary 3.4 to derive the following basic regularity information on the fluid component.
Again using (3.22), we thus obtain that with λ := 1 for all t ∈ (0, T max ), (3.23) so that since by (3.21), we firstly infer upon applying Lemma 3.5 that for all t ∈ (0, T max ). 4. An advanced quasi-entropy property: estimates in L p × W 1,2q × W 1,2 for some p > 2 and q > 1 This section will be devoted to the derivation of some time-independent integrability properties for n, and especially also for the signal gradient ∇c, in some superquadratic Lebesgue spaces. This will be achieved in Lemma 4.8 by analyzing a functional which extends that from (1.7) through an additional summand Ω |∇u| 2 , and in the course of an appropriate estimation of the respective contributions stemming from the interaction mechanisms in (1.3) we shall make essential use the regularity features of u asserted by Lemma 3.6 together with the L r bounds for c from Lemma 3.3.

Interpolating between zero-order and weighted second-order expressions
In order to prepare a suitable exploitation of this latter information currently available for c, let us briefly derive a functional inequality which interpolates gradients between weighted L 2 norms of Hessians and some zero-order Lebesgue norms. Though being quite in the spirit of precedents that involve L ∞ norms of the considered functions ( [40, Lemma 3.8]), their particular preciousness in the context of our present purposes originates from their meaningful applicability already in situations when L r bounds are a priori known for some finite r; in contrast to a previously obtained variant of a similar flavor ( [36, Lemma 2.7]), our result does not involve any additional zero-order summands on its right-hand side, and moreover is fully homogeneous with respect to multiplication of the estimated function by constant factors. Lemma 4.1. Let q > 1 and λ > 2 be such that λ < 2q + 2. Then there exists C = C(q, λ) > 0 such that for all ϕ ∈ C 2 (Ω) fulfilling ∂ϕ ∂ν = 0 on ∂Ω, Proof. Since ∂ϕ ∂ν = 0 on ∂Ω, an integration by parts followed by an application of the Cauchy-Schwarz inequality shows that since |Δϕ| ≤ √ 2|D 2 ϕ| in Ω, Keller-Segel-Navier-Stokes system Page 13 of 24 72 Here we may rely on our assumption λ < 2q + 2 to infer that by the Hölder inequality, , and that thus (4.1) holds with C(q, λ) : As a particular implication specifically adapted to our context, let us note the following.

4.2.
Analyzing Ω n p + Ω |∇c| 2q + Ω |∇u| 2 for some p > 2 and q > 1 Let us now turn our attention to the major part of this section by noticing the preliminary outcomes of three testing procedures applied to (1.3). For convenience in notation, we may and will assume here that the exponent α in (1.2) be such that α < 1. Assume (1.2) with some K f > 0 and α ∈ (0, 1), and let p > 2 and q > 2 − α. Then there

Lemma 4.3.
and Proof. In view of the first equation from (1.3), using (1.2) along with our assumption that α < 1 we see that due to Young's inequality, Here we note that our hypothesis p > 2 ensures that θ : whence a further application of Young's inequality shows that for all t ∈ (0, T max ). (4.8) Observing that pθ = p(q−1)−2 (1−α) q+α−2 and that we thus infer (4.4) from (4.8) when inserted into (4.7). To derive (4.5), we use that according to an argument due to [20,Lemma 3.2] we can find to see intergating by parts in the second equation from (1.3) that since ∇c · ∇Δc = 1 2 Δ|∇c| 2 − |D 2 c| 2 and |∇c| ≤ √ because for all t ∈ (0, T max ), by solenoidality of u. Since Young's inequality asserts that for all t ∈ (0, T max ), and that here due to the fact that λ : satisfies λ > θ > 1 we have from (4.9) we readily obtain (4.5) upon noticing that 2q ≥ 1 and computing Finally, testing the Navier-Stokes subsystem of (1.3) by Au we find that thanks to Young's inequality, for all t ∈ (0, T max ), where on the basis of our assumption on f we may clearly invoke Lemma 3.6 to infer finiteness of sup t∈(0,Tmax) Ω |u(·, t)| 2 , and to thus see by means of a Gagliardo-Nirenberg interpolation and Young's inequality that with some C 2 > 0 and C 3 > 0 we have for all t ∈ (0, T max ).
Next intending to estimate the expressions on the right-hand sides in (4.4), (4.5) and (4.6) through appropriate interpolation, we shall first focus on the integrals in (4.4) and (4.5) that involve n. By merely resorting to mass conservation as the only time-independent information on n currently at hand, we see that these can favorably be controlled by means of the corresponding dissipation rate in (4.4) when the exponent q is suitably large relative to p: Then for all η > 0 there exists C(η) = C(η; p, q) > 0 such that Proof. We first note that (4.12) together with our assumption p > 2 implies that q > (1−α)·2+α = 2−α, and that thus also is well-defined with θ > 1. Therefore, the Gagliardo-Nirenberg inequality in conjunction with (2.2) yields C 1 = C 1 (p, q) > 0 and C 2 = C 2 (p, q) > 0 such that so that since (4.12) furthermore ensures that and that thus 2(θ−1) p < 2, an application of Young's inequality readily yields the claim.
On the other hand, requiring q not to be too large as compared to p enables us to directly apply Corollary 4.2 in order to similarly estimate the integrals merely involving the taxis gradient against the dissipated quantity in (4.5).
Apart from that, let us prepare our subsequent estimation of the rightmost summand in (4.5) by the following two lemmata, the first of which again quite trivially results from Corollary 4.2 when q is suitably small. Lemma 4.6. Assume (1.2) with some K f > 0 and α > 0, and let q ≥ 1 be such that Then for all η > 0 there exists C(η) = C(η; q) > 0 with the property that Proof. This immediately follows from Corollary 4.2, because (4.16) warrants that 2q−(2q+2) = q−2 < 0.
Secondly, the corresponding fluid velocity gradient will be controlled by making use of the following result that relies on the L 2 bound for u provided by Lemma 3.6. Proof. Since well-known regularity features of the Stokes operator guarantee that A(·) L 2 (Ω) defines a norm equivalent to · W 2,2 (Ω) on D(A) ( [31]), by means of the Gagliardo-Nirenberg inequality we see that with some C 1 > 0 and C 2 > 0 we have Due to the boundedness property in (3.19) this already establishes (4.18).
It now turns out that if one last time we adequately make explicit use of our overall assumption on validity of (1.2) with some positive α, then we can find p > 2 and q > 1 in such a way that the assumptions (4.12), (4.14) and (4.16) are simultaneously satisfied. We may therefore derive the following from a combination of Lemma 4.3 with Lemmas 4.4-4.7. and as well as Proof. Assuming without loss of generality that α < 1, we fix any q ∈ (2 − α, 2) and use that then in choosing p ∈ (q + α, q−α 1−α ). Then necessarily p > (2 − α) + α = 2, so that Lemma 4.3 applies so as to yield C 1 > 0 such that d dt for all t ∈ (0, T max ). To derive (4.19)-(4.21) from this, we employ Lemma 3.6 to fix C 2 > 0 such that , by using Young's inequality along with Lemma 4.4 we can find C 3 > 0 fulfilling Moreover, taking C 4 > 0 such that in accordance with Lemma 4.7 we have in view of Young's inequality we obtain that 2 , we thereafter combine Young's inequality with Lemmas 4.6 and 4.5 to infer the existence of C 7 > 0 satisfying so that since, again by Young's inequality, upon collecting (4.25), (4.27) and (4.28) we conclude from (4.22) that has the property that To proceed from this, we note that if T max ≤ 2 and hence τ = 1 2 T max , then (4.23) implies that and that in the case when T max > 2 and thus τ = 1, for eack positive integer N and any k ∈ {0, . . . , N −1} we similarly obtain from (4.23) that meaning that for any such N and k, Therefore, in both these cases we find that so that an integration of (4.29) shows that due to (4.24), for all t ∈ (0, T max ).
Along with the fact that in (4.19) and (4.20) we have p > 2 and 2q > 2, the latter now provides sufficient information on regularity in the cross-diffusive and transport-related contributions to the first equation from (1.3) to ensure boundedness of n actually with respect to the spatial L ∞ norm. Proof. Using (1.2) and the fact that D(A ϑ ) → L ∞ (Ω) for all ϑ ∈ ( 1 2 , 1) ( [14,17]), on the basis of (4.19), (4.20) and (5.1) we readily infer the existence of r > 2 and C 1 > 0 such that h := f (|∇c| 2 )∇c + nu satisfies h(·, t) L r (Ω) ≤ C 1 for all t ∈ (0, T max ).

(5.3)
We then take any λ ∈ (2, r) and employ the comparison principle along with a known smoothing estimate for the Neumann heat semigroup (e tΔ ) t≥0 on Ω ( [12]) to see that with some C 2 > 0 and μ > 0 we have We have thereby collected all ingredients needed to derive our main result from Lemma 2.1.
proof of Theorem 1.1. We only need to combine the bounds provided by Lemma 5.2, (4.20) and Lemma 5.1 with Lemma 2.1.