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Global weak solutions to a strongly degenerate haptotaxis model

We consider a one-dimensional version of a model obtained in [C. Engwer, A. Hunt, and C. Surulescu: Effective equations for anisotropic glioma spread with proliferation: a multiscale approach and comparisons with previous settings, IMA J. Math. Med. Biol. (2015), doi: 10.1093/imammb/dqv030] and describing the anisotropic spread of tumor cells in a tissue network. The model consists of a reaction-diffusion-taxis equation for the density of tumor cells coupled with an ODE for the density of tissue fibers and allows for strong degeneracy both in the diffusion and the haptotaxis terms. In this setting we prove the global existence of weak solutions to an associated no-flux initial-boundary value problem.


Introduction
Models with degenerate diffusion in the context of taxis equations have received increased interest during the last decade. They describe the dynamics of a cell population in response to a chemoattractant [4,9,14], moving up the gradient of an insoluble signal (haptotaxis) [17], or performing both chemo-and haptotaxis [10,13,16]. In this work we consider a reaction-diffusion-transport-haptotaxis model which is inspired by the effective equations obtained in [6] via parabolic scaling upon starting from a multiscale model for glioma invasion in the anisotropic brain tissue and relying on the setting introduced in [5]. More precisely, the following PDE-ODE model was considered for the density function p(t, x, v, y) of glioma cells depending on time t, position x ∈ R n , velocity v ∈ V := sS n−1 , and density y ∈ Y := (0, R 0 ) of cell surface receptors 1 bound to tissue fibers, and for the subcellular dynamics simplified to mass action kinetics of the mentioned receptor binding: ∂ t p + ∇ x · (vp) + ∇ y · (G(y, w)p) = L[λ]p + P(p) (1.1) y = G(y, w).
(1.2) Thereby, w(x) represents the (macroscopic) volume fraction of tissue, the turning operator L[λ]p := −λ(y)p + V λ(y)K(x, v, v ′ )p(v ′ )dv ′ describes the reorientation of cells due to contact guidance by 1 tissue, and the term P(p) := µ(x,p, v) Y χ(x, y, y ′ )p(t, x, v, y ′ )w(x)dy ′ models proliferation subsequent to cell-tissue interactions. The function λ(y) denotes the cell reorientation rate, K(x, v, v ′ ) is the turning kernel depending on the directional distribution q(x, v) of tissue fibers (obtained from diffusion tensor imaging data), µ represents the proliferation rate depending on the macroscopic cell densitȳ p = V Y p(t, x, v, y)dydv, and χ is a kernel characterizing the transition from the state y to the state y ′ during a proliferative action.
An appropriate parabolic scaling led to the macroscopic equation for (an approximation of) the tumor cell density: ∂ t u − ∇∇ : (D T u) + ∇ · (a(w)D T ∇w u) = wµ(x, u)u, where a(w) is a function containing both macroscopic and subcellular level information, D T = const V qv ⊗ vdv is the tumor diffusion tensor encrypting the medical data about the structure of brain tissue, and ∇∇ : (D T u) = ∇ · (D T (x)∇u) + ∇ · (ζ(x)u) (1.4) with the drift velocity ζ(x) = const V v ⊗ v∇qdv. For more details and the precise definitions we refer to [6].
Equation (1.3) is of the reaction-diffusion-transport-(hapto)taxis type and characterizes the evolution of the tumor cell density for a known underlying structure of brain tissue; in practice, the functions q and w are assessed at a certain time point t from medical data. This facilitates both its mathematical analysis and efficient numerical handling, however in fact the tumor evolution in a patient also induces dynamical changes in the tissue such as e.g. depletion or remodeling, which play an essential role in the disease development, see e.g. [2,12] and the references therein. Therefore, a further equation is needed to describe these tissue modifications under the influence of tumor cells. Although in practice it is not feasible from the viewpoint of medical imaging to assess the tissue structure dynamically, by way of model-based predictions relying on such PDE-ODE coupled systems it is possible to use a sequence of just a few images in order to obtain via numerical simulations a good approximation of the dynamics over the whole timespan of interest.
Another issue is related to possible (local) degeneracies of the tumor diffusion tensor D T (x), which is particularly relevant e.g. when modeling resected or irradiated regions of the tumor, where the tissue has been depleted as well. In the respective domains, this indeed reduces the otherwise diffusiondominated PDE (1.3) to a hyperbolic transport equation with nonlinear source term. The mathematically quite delicate features of such strongly degenerate systems become manifest already in the case when any taxis or source terms are absent, that is, when a ≡ 0 and µ ≡ 0 in (1.3). Indeed, in [7] the linear scalar parabolic equation has been studied, motivated among others by a monoscale model for anisotropic glioma spread in [11], and it was shown there that if the functions d 1 and d 2 are smooth and nonnegative and such that d 1 is strictly positive but d 2 vanishes precisely in some subinterval [a, b] of (0, L y ), then solutions to an associated no-flux initial-boundary value problem asymptotically approach a singular state reflecting concentration of mass within the degeneracy region [0, L x ] × [a, b] and extinction outside.
In this paper we intend to provide a first step toward a mathematical understanding of corresponding sytstems when beyond such strongly degenerate diffusion processes, further crucial mechanisms and especially nonlinear haptotaxis are involved. In order to concentrate on essential aspects of such types of interplay within the framework of a model that captures the essential properties but beyond that remains as simple as possible, we may restrict to the spatially one-dimensional case, in which the tumor diffusion tensor D T in (1.3) actually reduces to a scalar function. In the context of a simple evolution law for the haptotactic attractant, particularly neglecting remodeling mechanisms, this leads to coupled parabolic-ODE systems of the form with given nonnegative functions d, ψ and h.
Although in our current 1D setting (1.6) the model in [6] loses most of its anisotropy relevance, some of it is retained in the space-dependent diffusion and haptotactic sensitivity coefficients. Likewise, the multiscality considered in [6] and leading to a haptotactic coefficient depending on the subcellular dynamics can still be partially retained in this model, in spite of the modified transport term, in which the drift velocity has now a simpler form, yet depending on d(x). The very presence of the haptotaxis term is a consequence of taking the receptor binding dynamics into account when describing the evolution of the cell density function on the mesoscopic level and scaling up to the macroscopic one. Hence, essential features of the model obtained in [6] are preserved even in this simplified, dimensionreduced setting.
Another related model featuring degenerate diffusion in the context of haptotaxis was proposed and investigated in [17]. The kind of degeneracy considered there is, however, different from the one in this and previous models, as it affects both the diffusion and the haptotaxis coefficients, thereby allowing the diffusion to degenerate due to one or both solution components (tumor cell density and tissue density). Unlike the present model, in [17] there is (apart from the taxis) no other transport term.

Problem setup and main result.
In order to make the essential mathematical aspects of (1.6) more transparent, let us write (1.6) in a form involving a constant haptotacitc sensitivity, which according to the simple ODE structure of the second equation therein can readily be achieved on with given parameter functions d : and with prescribed initial data u 0 and w 0 which are such that 0 ≤ u 0 ∈ C 0 (Ω) satisfies u 0 ≡ 0 and 0 ≤ w 0 ∈ W 1,2 (Ω) has the property that Ω (1.9) As for the parameter functions in (1.7), throughout our analysis we shall furthermore assume that and Beyond the analytically simplest case obtained on letting g(w) = w, w ≥ 0, this inter alia includes more general choices such as upon which via the substitution w = v 1+v , on the set of solutions fulfilling v < 1 the system (1.7) becomes formally equivalent to a corresponding initial-boundary value problem for the special version of (1.6), as proposed in [17] for modeling tumor invasion in a tissue network, thereby paying increased attention to the form of the haptotaxis coefficient. Specifically, the latter accounts for microscopic cell-tissue interactions, which -besides having a haptotaxis term at all-retains a supplementary trace of multiscality in our macroscopic model, although in a rather indirect way, as we do not explicitly couple some ODE for receptor binding kinetics to the two PDEs for u and v. The presence of d(x) in both diffusion/transport and haptotaxis coefficients is motivated by the deduction in [6].
The main results of our analysis indicate that even in this general setting, thus allowing for virtually arbitrary strength of degeneracies in diffusion, haptotactic cross-diffusion does not result in a finitetime collapse of solutions into e.g. persistent Dirac-type singularities. More precisely, let us introduce the following solution concept to pursued below, in which we use the abbreviation {d > 0} := x ∈ Ω d(x) > 0 which along with a corresponding definition of {d = 0} will frequently be used throughout the sequel.
Within this framework, a global solution of (1.7) can always be constructed: This paper is organized as follows: In Section 2 we introduce a regularized version of the degenerate problem, for which some useful properties are obtained. Section 3 is concerned with studying an entropy functional which allows to deduce a quasi-dissipative property of the regularized system, inter alia asserting global existence of its solution. Some precompactness and regularity properties of terms involved in that system follow in Sections 4 and 5, respectively, succeeded in Section 6 by regularity features of corresponding time derivatives. Sections 7 and 8 provide convergence properties of the approximate solution in the region with no degeneracy; further properties of the respective limits are obtained in Section 9. Finally, Section 10 concludes the existence proof for the strongly degenerate problem (1.7).

Regularized problems and their basic properties
In order to prepare the construction of an appropriate family of non-degenerate approximations of (1.7), according to the nonnegativity of d and the inclusion √ d ∈ W 1,∞ (Ω) we may first choose (d ε ) ε∈(0,1) ⊂ C 3 (Ω) in such a way that d εx = 0 on ∂Ω and that with some as well as and such that moreover and We next note that according to (1.12) it is possible to fix ε 0 ∈ (0, 1) such that g(M ) > ε 0 , whereupon with δ as introduced in the course of the definition (1.11) of M , for each ε ∈ (0, ε 0 ) we can choose and such that moreover δ ε → 0 as ε ց 0. It is then easy to see that one can find (η ε ) ε∈(0,ε 0 ) ⊂ (0, 1) with the two properties that indeed, it can readily be checked that this can be achieved on choosing with some suitably large A > 0. For ε ∈ (0, ε 0 ), we then let and consider the regularized variant of (1.7) given by Due to the additionally introduced artificial diffusion in the equation for w ε each of these problems can be viewed as a variant of the well-studied Keller-Segel chemotaxis system; in fact, as can be seen by straightforward adaptation of arguments well-established in the analysis of chemotaxis problems ( [1], [8], [15]), all these problems allow for local-in-time classical solutions which enjoy a favorable extensibility criterion: which solve (2.9) in the classical sense in Ω × (0, T max,ε ), and which are such that Let us first collect some basic properties of these solutions. We first assert some useful pointwise upper and lower bounds for w ε .
Proof. Since according to our choices of δ ε and M we have the inequality in (2.11) immediately results from the maximum principle applied to the second equation in (2.9). As a consequence thereof, in view of (1.12) we know that g ′ (w ε ) ≥ 0 in Ω×(0, T max,ε ), whence so that using (2.9) and (1.14) we see that and ∂w ∂ν = 0 on ∂Ω × (0, ∞) as well as for all x ∈ Ω by (2.8), the comparison principle therefeore ensures that w ε ≥ w in Ω × (0, T max,ε ) and that thus also (2.12) is valid.
Using the latter along with (1.10), we easily obtain the following information on the evolution of Ω u ε .

Lemma 2.3
With M as defined in (1.11) and ρ taken from (1.10), we have and Proof. We integrate the first equation in (2.9) and use (1.10) together with (2.11) to find that and that hence (2.13) holds, from which in turn (2.14) results upon integration in time.

Implications of an entropy-like structure
Now the core of our approach consists in the detection of a favorable quasi-dissipative property of the system (2.9) which can be revealed by following the well-established strategy of considering the time evolution of a functional that combines a logarithmic entropy of the cell distribution with a properly chosen summand annihilating the correspondingly obtained cross-diffusive interaction integral. In order to clarify which precise form the latter takes in the context of the approximate problems (2.9), let us begin by separately tracking the logarithmic entropy.
Proof. Since u ε is positive inΩ × (0, T max,ε ) by the strong maximum principle, we may multiply the first equation in (2.9) by ln u ε and integrate by parts to see that Here by Lemma 2.3 we have and using Young's inequality and (2.2) we obtain Thanks to a favorable exact relationship between the approximate signal absorption rate 0 ≤ u → u 1+ηεu and the tactic sensitivity 0 ≤ 1 (1+ηεu) 2 in (2.9), an exact compensation of the first summand on the right of (3.1) can be achieved on complementing the above by the following.

Proof.
Using that g(w ε ) is positive inΩ × [0, T max,ε ) due to Lemma 2.2 and (1.12), on the basis of the second equation in (2.9) we compute Here we expand We next use the identity so that on the right-hand side of (3.6) we can employ Young's inequality to see that for all t ∈ (0, T max,ε ), again due to (2.2). In conjunction with (3.7) and (3.6) this yields (3.5).
In fact, combining the latter two lemmata yields a quasi-entropy inequality, the essential implications of which can be summarized as follows.

13)
where with T max,ε as in Lemma 2.1 we have set T ε := min{T, T max,ε }.
As one consequence of (3.25) when combined with the boundedness of uε (1+ηεuε) 2 and the uniform positivity of d ε , we can infer that in fact our approximate solutions cannot blow up in finite time: Lemma 3.5 For each ε ∈ (0, ε 0 ), the solution of (2.9) is global in time; that is, in Lemma 2.1 we have T max,ε = ∞.
In view of the extensibility criterion (2.10), together with (3.27) and (3.28) this shows that our assumption T max,ε < ∞ was absurd.
4 Weak precompactness properties of u ε f (x, u ε , w ε ) and u ε g(w ε ) in L 1 In appropriately passing to the limit in the zero-order integrals appearing in the respective weak formulations of (2.9), we shall make essential use of two compactness properties of the solutions thereof which appear to go beyond trivial implications of the bounds provided by Lemma 3.3. As a preparation for our arguments in this respect, let us state the following observation on a lower bound for all possible values of u ≥ 0 at which u·f − (x, u, w) may become large for some x ∈Ω and w ∈ [0, M ]. This will be used in Lemma 4.2 to assert that for arbitrarily large κ > 0 one can pick N > 0 in such a way that whenever u ε f − (·u ε , w ε ) ≥ N , we know that u ε ≥ κ. and where passing to a subsequence if necessary we may assume that as N → ∞ we have By continuity of f − , however, this would imply that and thereby contradict (4.4).
Making use of the latter, by means of the Dunford-Pettis theorem we can now establish suitable compactness properties of the rightmost summands in the first two equations in (2.9).

Regularity properties of
In order to further prepare our limit procedure, especially with regard to pointwise convergence of u ε and of convergence in the cross-diffusive flux term d ε uε (1+ηεuε) 2 w εx in (2.9), we next plan to combine the weak compactness feature of the part √ d ε w εx thereof, as naturally implied by Corollary 3.4, by an appropriate result on convergence in the complementary factor √ d ε uε (1+ηεuε) 2 in a strong L 2 topology. To achieve this in Lemma 8.1 by using underway an argument based on the Aubin-Lions lemma, let us suitably interpolate between the inequalities in (2.14) and (3.11) to derive the following spatio-temporal estimates for the quantity √ d ε u ε forming the core of the factor in question.

Regularity in time
As a final preparation for our first subsequence extraction procedure, we combine our previously gained estimates to obtain some regularity properties involving time derivatives of the solution components u ε and w ε .
Therefore, in view of Lemma 6.1 the Aubin-Lions lemma asserts that for any such T , is relatively compact in L 2 ((0, T ); L 2 (Ω)), from which it follows by a standard argument that for a suitable sequence (ε k ) k∈N ⊂ (0, ε 0 ) and some z ∈ L 2 loc ([0, ∞); L 2 (Ω)) we have ε k ց 0 as k → ∞ and and as ε = ε k ց 0. Since d ε → d a.e. in Ω as ε ց 0 by (2.3), this means that if we let u(x, t) := z 2 (x,t) d(x) − 1 for x ∈ {d > 0} and t > 0, then (7.7) and (7.6) imply (7.1) and (7.3), whereupon (7.1) a posteriori also shows that u must be nonnegative. We next make use of the estimate (3.9) from Lemma 3.3 to infer that for T > 0 and ε ⋆ (T ) as above, is bounded in L 1 (Ω × (0, T )), so that the Dunford-Pettis theorem guarantees that (u ε ) ε∈(0,ε⋆(T )) is relatively compact with respect to the weak topology in L 1 (Ω × (0, T )), and that hence (7.2) can be achieved on extracting a subsequence 9 Solution properties of u and w We are now ready to make sure that ( u, w) indeed solves (1.7) when restricted to {d > 0} in the following sense.
Lemma 9.1 Let u and w be as obtained in Lemma 7.1.
10 Proof of Theorem 1.2 We can finally extend the above spatially local solution in an evident manner so as to become a global weak solution in the flavor of Definition 1.1. In the verification of the desired solution property near the boundary of {d > 0} we shall make use of the following consequence of the inclusion √ d ∈ W 1,∞ (Ω).
By means of an appropriate cut-off procedure we can thereby proceed to show that the natural extension of ( u, w), consisting of a solution to the ODE system formally associated with (1.7) in {d = 0} indeed solves (1.7) in the desired sense.
Proof of Theorem 1.2. We let u and w denote the functions defined on {d > 0} × (0, ∞) in Lemma 7.1, and for fixed x ∈ {d = 0} we let ( u(x, ·), w(x, ·)) ∈ (C 1 ([0, ∞))) 2 be the solution of the initial-value problem In order to verify (1.20), we first make use of the fact that by continuity of d the set {d > 0} is relatively open inΩ, and hence consists of countably many connected components; that is, there exist an index set I ⊂ N and intervals P i ⊂Ω, i ∈ I, such that {d > 0} = i∈I P i and P i ∩ P j = ∅ for i, j ∈ I with i = j. Now for each i ∈ I, there exist a i ∈Ω and b i ∈Ω such that (a i , b i ) ⊂ P i ⊂ [a i , b i ], where a i ∈ P i if and only if a i ∈ ∂Ω and b i ∈ P i if and only if b i ∈ ∂Ω. For fixed δ ∈ (0, 1), defining δ i := 2 −i δ, i ∈ I, it is then possible to pick (ζ (i) δ ) i∈I ⊂ C ∞ (Ω) such that for all i ∈ I we have 0 ≤ ζ (i) δ ≤ 1 inΩ, ζ (i) δ (x) = 1 whenever x ∈ P i is such that dist (x, ∂P i ) ≥ δ i , ζ where in the exceptional case a i ∈ ∂Ω we can additionally achieve that ζ (i) δ ≡ 1 holds even throughout [a i , b i − δ i ], and where, similarly, in the case b i ∈ ∂Ω we require that ζ Here we may use that 0 ≤ ζ δ ≤ 1 and that as δ ց 0 we have ζ δ → 1 a.e. in {d > 0} to infer from the dominated convergence theorem that The derivation of (1.21) is much less involved: Given ϕ ∈ C ∞ 0 (Ω × [0, ∞)), from (10.5) and (10.2) we first obtain that whereas with (ζ δ ) δ∈(0,1) as introduced above we obtain from Lemma 9.1 that for all δ ∈ (0, 1). Using that w and ug(w) belong to L 1 loc ([0, ∞); L 1 ({d > 0})) by Lemma 2.2, Lemma 4.2 and Lemma 7.1, we may again employ the dominated convergence theorem here to see that in the limit δ ց 0, (10.23) implies that