REFINED REGULARITY AND STABILIZATION PROPERTIES IN A DEGENERATE HAPTOTAXIS SYSTEM

. We consider the degenerate haptotaxis system endowed with no-ﬂux boundary conditions in a bounded open interval Ω ⊂ R . It was proposed as a basic model for haptotactic migration in heterogeneous environments. If the diﬀusion is degenerate in the sense that d is non-negative, has a non-empty zero set and satisﬁes (cid:82) Ω 1 d < ∞ , then it has been shown in [12] under appropriate assumptions on the initial data that the system has a global generalized solution satisfying in particular u ( · ,t ) (cid:42) µ ∞ d weakly in L 1 (Ω) as t → ∞ for some positive constant µ ∞ . We now prove that under the additional restriction (cid:82) Ω 1 d 2 < ∞ we have the strong convergence u ( · ,t ) → µ ∞ d in L p (Ω) as t → ∞ for any p ∈ (1 , 2). In addition, with the same restriction on d we obtain improved regularity properties of u , for instance du ∈ L ∞ ((0 , ∞ ); L p (Ω)) for any p ∈ (1 , ∞ ).


1.
Introduction. Tumor invasion into the healthy tissue relies on a plethora of processes. However, many types of cancer cells are only able to move if they adhere to the tissue fibers in the extracellular matrix. Hence, they migrate from places with low densities of the tissue fibers (and corresponding adhesive molecules on the fibers) to places with higher densities. This process is called haptotaxis (see e.g. [4]) and has been present in a growing number of macroscopic models for tumor invasion into the tissue (see e.g. [5] for one of the first models). Consequently, the mathematical analysis of haptotaxis systems has got growing interest during the past decade. Mathematically, these systems usually consist of a cross-diffusive parabolic PDE for the tumor cell density (modeling diffusion and haptotaxis) coupled with an ODE for the density of the tissue fibers (since the latter is a non-diffusive attractant for the tumor cells). In most of these systems the random movement of cancer cells is described by non-degenerate diffusion of Fickian type (see e.g. [1,Section 4.3] for a recent survey), only few containing degenerate diffusion (see e.g. [13]). However, in organs with very heterogeneous tissue (e.g. in the brain) recent modeling approaches suggest that non-Fickian diffusion operators could possibly be more adequate, among them the so-called myopic diffusion (see e.g. [2]).
Finally, we choose initial data satisfying u 0 , w 0 ∈ C 0 (Ω), u 0 ≥ 0, u 0 ≡ 0, w 0 ≥ 0, √ w 0 ∈ W 1,2 (Ω) and Ω d 2 In [12] the above assumptions were prescribed, but with Ω 1 d < ∞ instead of (2), and it was shown that there exists a global generalized solution (u, w) to (1) in the sense of [12,Definition 2.1] , u obeys conservation of mass and the solution has the asymptotic behavior w(·, t) → 0 in L ∞ (Ω) as well as where It is the purpose of the present paper to establish a strong convergence of u(·, t) to µ∞ d in some space L p (Ω). To this end, it turns out that instead of requiring 1 d belonging to L 1 (Ω) we need that it belongs also to L 2 (Ω). The latter means an additional restriction of the behavior of d near its zeros as compared to the setting from [12].
Main results. By requiring the generalization (6) of (2) we have the following main results: Let Ω ⊂ R be a bounded interval, and suppose that d ∈ C 0 (Ω) ∩ C 1 ({d > 0}) is nonnegative and such that Moreover, let g ∈ C 2 ([0, ∞)) be such that g(0) = 0 and that (3) is valid with some γ > 0 and γ > 0, and assume that the initial data u 0 and w 0 satisfy (4). Then the global generalized solution (u, w) of (1) from [12, Theorem 1.1] has the additional properties that and and furthermore with µ ∞ := Ω u0 While the proof of the regularity properties in [12] is mainly based on bounds obtained from an energy-like inequality for regularized approximations of (1), our approach to prove (7) stems from the observation that the flux term in the first equation of (1) has the form x . This idea was established in non-degenerate haptotaxis systems in [6,7] and leads for a supposedly given smooth solution of (1) to the identity d dt Ω . In order to rigorously prove an appropriate ODI for a regularized approximation of p e w , in the regularized version of the above identity we estimate the last term on the right-hand side, where u appears at a high power, by using on the one hand an interpolation inequality of Gagliardo-Nirenberg type which may be viewed as a derivate of an observation originally made in [3] and on the other hand estimates provided by [12] for the approximate problems (11), see (12)-(23) below.
Here we will make essential use of our overall assumption that 1 d does not only belong to L 1 (Ω) but even to L 2 (Ω). In addition, we will also have to adequately cope with terms stemming from the artificial diffusion introduced in the second equation in (11), in this context no longer acting in a dissipative manner, and this will be achieved by substantially relying on the boundedness properties from [12]. These ingredients will then lead to uniform L p estimates for the regularizations d ε u ε of du. This we will do in Section 2.1, after having stated the approximate problems (11) along with some of their important properties from [12] in the beginning of Section 2.
If in addition to the assumptions from Theorem 1.1 we require w0 d ∈ L ∞ (Ω), then [12, Theorem 1.3] implies for a.e. t > 0 the existence of positive constants C 1 (t) and C 2 (t) such that for a.e. x ∈ Ω.
Hence, we cannot expect to achieve bounds for u ε itself in L p (Ω) for large p. However, for any compact set K ⊂ {d > 0} we obtain uniform bounds for u ε in L ∞ (K). This is done in Section 2.2 with the help of a transformation to an inhomogeneous heat equation and the use of well-known estimates for the heat semigroup. These bounds in conjunction with standard parabolic regularity results then yield uniform interior Hölder estimates for u ε in domains of the form K × (τ, 1 √ ε ), see Section 2.3. As all these estimates are uniform with respect to ε, by taking the limit ε 0 along an appropriate subsequence we finally conclude in Section 3 the properties of u claimed in Theorem 1.1. A short appendix contains the proof of the announced interpolation inequality of Gagliardo-Nirenberg type.
2. Refined uniform regularity properties for approximating problems. As in [12] we consider the approximating problems x ∈ Ω, t > 0, for ε ∈ (ε j ) j∈N , where (ε j ) j∈N ⊂ (0, 1) with ε j 0 as j → ∞ as well as d ε and w 0ε are defined in [12, Lemma 2.2 and Lemma 2.6]. In view of the latter references, these functions have the following properties, which we will frequently use in the sequel: For all ε, ε ∈ (ε j ) j∈N we have where w 0j ∈ L ∞ (Ω) is nonnegative and in particular satisfies √ w 0j ∈ W 1,2 (Ω) as well as supp w 0j ⊂ {d > 0} and, as j → ∞, w 0j w 0 in Ω. Furthermore, it was shown in [12, Section 4] that for any ε ∈ (ε j ) j∈N there is a global classical solution (u ε , w ε ) to (11). According to [12, Lemmas 2.7., 2.8 and 3.5], for any ε ∈ (ε j ) j∈N this solution fulfills and there exists a constant C > 0 such that for any ε ∈ (ε j ) j∈N we have 2.
1. An estimate for d ε u ε in L p (Ω). A crucial step for our asymptotic analysis consists in deriving appropriate ε-independent regularity properties of the solution component u ε in Lebesgue spaces involving higher integrability powers. In order to prove the desired L p estimate for d ε u ε for abitrary large finite p, we will rely on an interpolation using Lemma 4.1 as well as the estimates from [12] and our assumption that 1 d belongs to L 2 (Ω) and not only to L 1 (Ω) as described in the introduction.
Proof. Let us first follow an idea well-established in related non-degenerate frameworks ( [6,7]) to rewrite the flux in the first equation in (11) according to This, namely, suggests to test the PDE in question by (d ε u ε e −wε ) p−1 to obtain where the second summand on the right can be expanded so as to yield Here by Young's inequality, (20) and (3), by (17), and since (22) warrants that with some c 1 > 0 we have we infer that for any choice of ε ∈ (ε j ) j∈N ⊂ (0, 1) and all t > 0, where .
Similarly, the second last summand in (26) can be controlled according to , whereas for the rightmost term in (26) we find on invoking the Cauchy-Schwarz inequality that In summary, (26)-(29) show that writing c 5 := c 2 + c 3 + c 4 we obtain for all t > 0, and in order to further estimate the last summand herein, employing the Gagliardo-Nirenberg inequality we pick c 6 > 0 such that for all ϕ ∈ W 1,2 (Ω) and invoke Young's inequality in fixing c 7 > 0 such that for all a ≥ 0 and b ≥ 0.
Using that by (16) and (19) we know that we thereby see that for all t > 0, with c 9 := c p+1 5 c p+1 6 c 7 c p 8 + c 5 c 6 c p 8 . On the right-hand side of (25), we next proceed to use (3) and (20) as well as (14) to estimate noting that c 10 := (p − 1)γM e 2M Ω 1 d 2 is finite thanks to our overall assumption on square integrability of 1 d . Here we recall that (21) provides c 11 > 0 such that for all t > 0, which by (16), (19) and (20) entails that there exists c 12 > 0 such that Therefore, applying Lemma 4.1 to q := 2 p and recalling (31) we see that with some c 13 ≥ p + 1 we have Since finally a Sobolev inequality associated with the embedding W 1,2 (Ω) → L ∞ (Ω) yields c 15 > 0 such that for all ϕ ∈ W 1,2 (Ω), again by means of (31) and (14), as Ω 1 d is finite we can find c 16 > 0 such that for all t > 0 and hence for all t > 0. Therefore, (34) shows that for for all t > 0 and thereby yields the claim, because by (16).

2.2.
A local L ∞ bound for u ε in {d > 0}. We first plan to derive a bound for u ε in L ∞ (K) with arbitrary compact K ⊂ {d > 0}, which in view of the equation satisfied by the quantity d ε u ε e −wε apparently does not follow from Lemma 2.1 in a trivial manner upon performing a straightforward Moser-type iteration. Fortunately, in the present one-dimensional situation an alternative approach can be based on a variable transformation which allows for a reduction to a linear inhomogeneous heat equation: Lemma 2.2. Let J ⊂ Ω be an interval and x 0 ∈ J, and given ε ∈ (ε j ) j∈N let as well as J ε := φ ε (J). Then φ ε ∈ C ∞ (J) is strictly increasing, and if for arbitrary ζ ∈ C 2 ( J ε ) we let the functions Z ε , W ε and H ε be definied on J ε × [0, ∞) by setting and as well as for y ∈ J ε and t ≥ 0, then and b (2) ε (y, t) := 2ζ y (y) + D ε (y)ζ(y) Z εy (y, t) (42) as well as b (3) ε (y, t) := − ζ yy (y) + ζ y (y)W εy (y, t) + D ε (y)ζ(y)W εy (y, t) Z ε (y, t) for (y, t) ∈ J ε × (0, ∞). Moreover, if J ∩ ∂Ω = ∅, and if for some ε ∈ (ε j ) j∈N we have ζ y = 0 on φ ε (J ∩ ∂Ω), then Proof. The claimed regularity and monotonicity properties of φ ε are evident from the inclusion d ε ∈ C ∞ (Ω) and the positivity of d ε on Ω, as asserted by (12) and (14). To verify (40), we only need to combine (35)-(39) in computing as well as and, similarly, so that by (11), . By furthermore using the identities and H εyy = ζZ εyy + 2ζ y Z εy + ζ yy Z ε , from this we readily derive (40). Finally, (45) is a direct consequence of (47) and the fact that due to (46), the boundary condition for u ε in (11) together with the property d εx | ∂Ω = 0 achieved in (15) warrants that Z εy = 0 on φ ε (J ∩ ∂Ω).
In consequence, deriving bounds of the desired type essentially reduces to suitably estimating the inhomogeneities in (40) on the basis of [12, Lemma 3.5] and Lemma 2.1. Indeed, this will form the core of the otherwise mainly technical reasoning in the following.
Now going back to (57), using (61), (63) and the fact that |ζ| ≤ 1, by the Hölder inequality we find that therein for each ε ∈ (ε j ) j∈N such that ε < ε (1) we have for all t > 0, and that hence in view of (57) there exists c 14 > 0 such that for all t > 0.

2.3.
Local Hölder regularity of u ε in {d > 0}. Now with the above boundedness information at hand, we may invoke standard parabolic regularity to obtain the announced interior Hölder regularity property. for all t > 0 and thereby, thanks to (6), already establishes (8). Secondly, given any 4. Appendix: A refined interpolation inequality. We prove the following interpolation inequality of Gagliardo-Nirenberg type which is based on an observation originally made in [3]. for all ϕ ∈ W 1,2 (Ω).