A CHEMOTAXIS-HAPTOTAXIS SYSTEM WITH HAPTOATTRACTANT REMODELING: BOUNDEDNESS ENFORCED BY MILD SATURATION OF SIGNAL PRODUCTION

. We consider the chemotaxis-haptotaxis system ) , in a bounded convex domain Ω ⊂ R n with smooth boundary, where χ,ξ,µ and η are positive constants, and where f ∈ C 1 ([0 , ∞ )) is a given function fulﬁlling f (0) ≥ 0 and f ( s ) ≤ K f ( s + 1) α for all s ≥ 0 with some K f > 0 and α > 0. It is asserted whenever 2 , the Neumann boundary problem with suitably regular initial data possesses a unique global and bounded

in a bounded convex domain Ω ⊂ R n with smooth boundary, where χ, ξ, µ and η are positive constants, and where f ∈ C 1 ([0, ∞)) is a given function fulfilling f (0) ≥ 0 and f (s) ≤ K f (s + 1) α for all s ≥ 0 with some K f > 0 and α > 0. It is asserted that whenever α < 1. Introduction. Taxis mechanism are known to play an important role in the process of cancer cell invasion of neighboring tissue. In addition to random motion, cancer cells are able to direct their migration toward the higher concentration of enzymes secreted by themselves, while they are moreover attracted by matrix molecules adhering to the tissue and consequently their movement is biased toward the higher density of tissue as well. Cancer cells usually undergo proliferation and death and compete surviving space with adjacent tissue. The diffusible enzymes can degrade the tissue, which is assumed to have a certain ability to recover to a normal level, and which competes for space with cancer cells.
A renowned macroscopic model accounting for these mechanisms, as proposed by Chaplain and Lolas ([4,5]), gives rise to studying the parabolic-parabolic-ODE initial-boundary value problem given by x ∈ Ω, t > 0, x ∈ Ω, t > 0, in a bounded domain Ω ⊂ R n with smooth boundary, and in fact a considerable literature has addressed questions from existence and qualitative theories therefor under various circumstances. In particular, quite a comprehensive knowledge could be generated in the case when η = 0, hence reflecting situations in which spontaneous tissue remodeling can be neglected. For such systems, various results on global existence of classical solutions are available under mild assumptions inter alia allowing for the most prototypical choice f (u) = u ( [18,19,24], cf. also [1] for a survey), and furthermore some results on boundedness and even on large time stabilization could be achieved, partially upon some additional assumptions on the initial data w 0 and on the model parameters µ and χ ( [10,24,27]), or upon some further simplification replacing the second identity in (1.1) by an elliptic equation ( [21,22]), or upon alternative model modifications by e.g. including nonlinear cell diffusion ( [13,31,26]). As for full versions of the system (1.1) which do contain tissue remodeling terms by admitting η to be positive, only little seems known, which from a mathematical perspective might be viewed as reflecting some additional destabilizing potential induced by the reaction term −ηuw, then entering (1.1) in a nontrivial manner, through possible effects on formation of sharp haptoattractant gradients ∇w which due to the lack of diffusion cannot expected to be regularized during evolution. Actually, in the case when f (u) = u the only results we are aware of in this field are either restricted to spatially two-dimensional settings, in which in fact a result on global existence and boundedness could recently be derived ( [16], cf. also the precedents [15] and [23]), or on the construction of weak solutions, which were indeed found to exist also in three-dimensional domains ( [17]), but the regularity information on which is yet rather poor and especially does not rule out the possibility of finite-time emergence of singularities.
Main results. The purpose of the present work consists in examining to which extent the regularity properties of the full system (1.1), thus including tissue remodeling, benefit from certain saturation effects in the production of enzymes at large densities of the tumor population. In particular, we shall subsequently consider (1.1), with positive parameters χ, ξ, µ and η, under that standing assumptions that and that the growth of f at large values of u is controlled from above according to with some K f > 0 and α > 0. Our hypotheses concerning the initial data will be that u 0 , v 0 and w 0 are nonnegative functions from C 2+ϑ (Ω) for some ϑ > 0, Addressing precisely this setup, for the physically relevant three-dimensional case a recent work ( [3]) has provided a result on global existence of bounded classical solutions under the condition that f grows at most in a considerably sublinear manner in the sense that in (1.3) we have α < 5 6 and hence The goal of this work is to develop an apparently novel type of mathematical approach, at its core based on an argument relying on well-known maximal Sobolev regularity estimates for inhomogeneous heat equations (see Section 4), which allows for a significant reduction of the gap between saturated and linear signal production expressed in (1.5), and our main results will reveal that (1.5) can actually be replaced with the assumption that merely when n = 3. More precisely, and more generally, we shall derive the following. the problem (1.1) admits a global classical solution (u, v, w) which is bounded in the sense that u(·, t) L ∞ (Ω) ≤ C for all t > 0 (1.8) with some C > 0.
2. Local existence and a convenient extensibility criterion. Following a rather standard reduction step (see e.g. [6,7,25] and [20] for some precedents), we conveniently transform (1.1) by the substitution a := e −ξw u, (2.1) through which, namely, in the framework of classical solutions the problem (1.1) becomes equivalent to (2.2)

YOUSHAN TAO AND MICHAEL WINKLER
In this setting, the following statement on local existence and extensibility can be derived by means of quite well-established arguments.
Lemma 2.1. Suppose that n ≥ 1 and that Ω ⊂ R n is a bounded domain with smooth boundary, that χ, ξ, µ and η are positive, and that (1.4) holds. Then there exist T max and a uniquely determined triple (a, w, v) of nonnegative functions a, v and w belonging to C 2,1 (Ω × [0, T max )) which form a classical solution of (2.2) in Ω × (0, T max ), and which are such that Furthermore, with u := e ξw a and τ := min{1, and t+τ t Ω as well as for all x ∈ Ω and t ∈ (0, T max ). (2.6) Proof. On the basis of a standard fixed point argument, the claims concerning local existence and extensibility can be verified by minor adaptation of well-documented reasonings, e.g. the proof detailed for a related two-dimensional analogue in [23, Lemmata 2.1 and 2.2], to the present multi-dimensional setting. The nonnegativity of a, v and w as well as (2.6) thereafter follow from the maximum principle, and since an integration in the first equation from (1.1) shows that due to the fact that Ω u 2 ≥ 1 |Ω| Ω u 2 by the Cauchy-Schwarz inequality we readily infer both (2.4) and (2.5).
Throughout the sequel, without further explicit mentioning we shall suppose that n ≥ 1 and Ω ⊂ R n is a bounded convex domain with smooth boundary, that (1.4) holds, that χ, ξ, µ and η are positive and µ > ξηM with M > 0 taken from (2.6), and let a, v, w and u as well as T max be as correspondingly provided by Lemma 2.1 and (2.1).
In order to conveniently relax the extensibility criterion (2.3), let us make sure that L ∞ bounds for a already entail spatial L q bounds for ∇w with arbitrarily large finite q, at least locally in time: Then for all p ≥ 2 and any T > 0 one can find C(p, T ) > 0 such that Proof. Rewriting the first equation in (2.2) in the form we see that thanks to (2.8), the coefficient functions as well as with some c 1 > 0 and c 2 > 0, because (2.8) clearly entails boundedness of v and ∇v in Ω × (0, T max ) through standard parabolic estimates (see e.g. [11,Lemma 4.1]). Therefore, by means of Young's inequality we see that with some [14]), and because |∆a| ≤ √ n|D 2 a|. We now note that thanks to the latter we may also infer from (2.8) that there exists Again using Young's inequality, we thus readily see that (2.11) implies that where maximal Sobolev regularity estimates ( [8]) along with (2.8) and (2.6) yield (2.14) As, independently, on testing the third equation in (2.2) by |∇w| p ∇w we easily find in view of (2.12) this entails that if we restrict t so as to satisfy t ∈ [0, t 0 ] with t 0 = t 0 (p) ∈ (0, min{T max , T }) fulfilling c 2 5 c 6 c 9 t 0 ≤ p 8 , then as a consequence of (2.13) we obtain that When combined with (2.12) and (2.15), this establishes (2.9) for any such t, and since t 0 could be chosen so as to depend only on p but not on, e.g., (a, v, w)(·, 0), we may repeat this procedure finitely many times if necessary to conclude that (2.9) actually holds for all t ∈ (0, min{T max , T }).
Thereby, (2.3) actually reduces to the following.
Lemma 2.3. The solution of (2.2) actually has the property that Proof. This directly follows by combining Lemma 2.1 with Lemma 2.2.
3. Implications of gradient estimates for v on integrability of a. In order to concretize our goal to be subsequently pursued, let us state the following observation which forms the starting point therefor, and which can be obtained by means of a standard testing procedure. The following lemma is the only place in this paper in which our standing assumption µ > ξηM is explicitly utilized.
In order to draw appropriate consequences of this, but also for later reference, let us recall from [30, Lemma 3.4] the following statement on control of the inhomogeneous part in a Duhamel formula associated with the ODI y (t) + λy(t) ≤ h(t) with given nonnegative h and λ > 0. Then for any choice of λ > 0, for all t ∈ (0, T ). Now estimating the crucial first integral on the right of (3.1) by means of two different strategies depending on whether n = 1 or n ≥ 2, using Lemma 3.2 we obtain from Lemma 3.1 the following criterion for the derivation of estimates for a which are of a similar flavor of those implied by (2.4) and (2.5), but which go beyond these when the parameter p addressed below satisfies p > 1.

YOUSHAN TAO AND MICHAEL WINKLER
Lemma 3.3. Suppose that p > 1 is such that with τ := min{1, 1 2 Then there exists C = C(p) > 0 such that Proof. According to Lemma 3.1, we can fix for all t ∈ (0, T max ), and in the case n ≥ 2 we use Young's inequality to see that with some c 3 = c 3 (p) > 0 we herein have Since clearly, by the same token and (2.6), as well as with some c 4 = c 4 (p) > 0 and c 5 = c 5 (p) > 0, from (3.5) we thus infer that in this case, for all t ∈ (0, T max ). Thanks to (3.2), on integration using Lemma 3.2 from this we readily obtain both (3.3) and (3.4).
In the case n = 1, in (3.5) we rather make use of the second summand on the left: By means of the Gagliardo-Nirenberg inequality, (2.4) and Young's inequality, namely, we see that with some positive constants c 6 = c 6 (p), c 7 = c 7 (p) and L 2 (Ω) + c 8 for all t ∈ (0, T max ).

A CHEMOTAXIS-HAPTOTAXIS SYSTEM 2055
As furthermore combining the Gagliardo-Nirenberg inequality with Young's inequality we easily find c 9 = c 9 (p) > 0 fulfilling 4. The core: gradually improving bounds for Ω a p and t+τ t Ω a p+1 . Inspired by the outcome of Lemma 3.3, our strategy now will consist in successively improving our knowledge on integrability of a, as measured through its norm in L ∞ ((0, T max ); L p (Ω)) and the expression sup t∈(0,Tmax−τ ) t+τ t Ω a p+1 for p ≥ 1, using the basic information from (2.4) and (2.5) as a starting point. More precisely, our goal will be to show unboundedness of the set S given by where again τ := min{1, 1 2 T max }, by means of a contradictory argument based on the fact that, as we shall see, the assumptions from Theorem 1.1 essentially warrant the existence of δ > 0 such that whenever p ∈ S, we also have p + δ ∈ S. In light of Lemma 3.3, the latter step amounts to developing the bounds for a implied by a supposedly valid inclusion p ∈ S into appropriate estimates for v and its gradient.
Though rather elementary, a first and quite fundamental observation relates such bounds for a to certain integrability properties of the inhomogeneity f (u) in the second equation from (1.1).
A first application thereof, here yet exclusively used for the choice σ = p+1 α in (4.2), entails a temporally uniform bound for v in a conveniently small L q space.
Lemma 4.2. Assume that α ∈ (0, 2), and suppose that p ≥ 1 is such that p ∈ S. Then there exists C(p) > 0 with the property that Proof. Integrating by parts in the second equation from (1.1), for t ∈ (0, T max ) we compute where the first summand on the right is nonpositive due to the fact that α < 2 implies that p + 1 − α ≥ 2 − α > 0. Since for the same reason we also have p+1 α > 1, we may furthermore invoke Young's inequality, which allows us to estimate the rightmost integral in (4.4) according to From (4.4) we therefore obtain that where the inclusion p ∈ S along with (2.6) ensures the existence of c 1 = c 1 (p) > 0 such that t+τ t h(s)ds ≤ c 1 for all t ∈ (0, T max − τ ), again with τ := min{1, 1 2 T max }. An application of Lemma 3.2 thus shows that and hence establishes (4.3).
Proof. Following a temporal localization procedure in the style of that e.g. from [29], we fix where according to our hypotheses on v 0 in (1.4), we see that with some Now since our assumption p > α − 1 ensures that p+1 α > 1, and that furthermore in both cases α ≤ 1 and α > 1 we also have σ ασ−p > 1, we may invoke a well-known result from maximal Sobolev regularity theory ( [8], [9]) to find c 2 = c 2 (p, σ) > 0 such that ≤ c 4 for all t ∈ (0, T max ), and that hence also for all t 0 ∈ [0, T max ), from (4.7) and (4.8) we obtain c 5 = c 5 (p, σ) > 0 such that for all t 0 ∈ [0, T max ).

4.2.
Temporally uniform W 1,q bounds for v via L p -L q estimates. An argument independent from that from Lemma 4.3, rather relying on direct application of smoothing estimates for the Neumann heat semigroup to a Duhamel representation of v, shows that Lemma 4.1 furthermore implies temporally uniform bounds for v in certain W 1,q spaces.
Upon optimizing the choice of σ here, we obtain a corresponding statement that will allow for somewhat more convenient handling in the sequel.
Next, if p > n 2 and p ≥ α, computing and observing that by means of (4.15) and a continuity argument we conclude that given any q ≥ 1 such that q < np (nα−p)+ we can pick σ ∈ J such that besides (4.10) also (4.9) holds, and that hence (4.14) again becomes a consequence of Lemma 4.4.
Likewise, if p ≤ n 2 is such that p > 2α − 1, then in view of (4.15) we choose σ = p+1 α to see that then any q < n(p+1) and that furthermore so that the claim can be derived from Lemma 4.4 also in this case.
Next, in the case when n 2 < p < nα we proceed quite similarly, first verifying that then (4.16) implies that nαr p+1 − n < np nα−p , and noting that (4.19) and (4.20) remain unchanged, whereas evidently np nα−p > 1 due to the fact that nα n+1 < n n+1 < 1 ≤ p. We can therefore fix a number q = q(p, r) which again has the properties that 1 ≤ q < r and that (4.23) holds, and which moreover satisfies q < np nα−p as well as (4.27) with b as accordingly defined through (4.26), the latter statement being valid due to (4.23). Consequently, we may argue precisely as in our derivation of (4.24) and (4.28) to infer that (4.17) is valid also in this case.
Finally, if p ≥ nα and r ∈ (1, ∞) is arbitrary, the claim directly results on applying Corollary 4.5 to q := r.
Our one-dimensional counterpart thereof, now choosing σ in Lemma 4.3 significantly smaller than before, reads as follows.