A new result for boundedness of solutions to a quasilinear higher-dimensional chemotaxis -- haptotaxis model with nonlinear diffusion

This paper deals with a boundary-value problem for a coupled quasilinear chemotaxis--haptotaxis model with nonlinear diffusion $$\left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)-\xi \nabla\cdot(u\nabla w)+\mu u(1-u-w),\\ v_t=\Delta v- v +u,\quad \\ w_t=- vw\\ \end{array}\right. $$ in $N$-dimensional smoothly bounded domains, where the parameters $\xi ,\chi>0$, $\mu>0$. The diffusivity $D(u)$ is assumed to satisfy $D(u)\geq C_{D}u^{m-1}$ for all $u>0$ with some $C_D>0$. Relying on a new energy inequality, in this paper, it is proved that under the conditions $$m>\frac{2N}{N+{{{\frac{(\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}} (\chi+\xi\|w_0\|_{L^\infty(\Omega)})}{(\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_0\|_{L^\infty(\Omega)})-\mu)_{+}}+1) (N+\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_0\|_{L^\infty(\Omega)})}{(\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}} (\chi+\xi\|w_0\|_{L^\infty(\Omega)})-\mu)_{+}}-1)}{N}}}}},$$ and proper regularity hypotheses on the initial data, the corresponding initial-boundary problem possesses at least one global bounded classical solution when $D(0)>0$ (the case of non-degenerate diffusion), while if, $D(0)\geq 0$ (the case of possibly degenerate diffusion), the existence of bounded weak solutions for system is shown. This extends some recent results by several authors.


Introduction
Cancer invasion is a very complex process which involves various biological mechanisms (see [2,18,6,7,10,15]). Chemotaxis is the oriented movement of cells along concentration gradients of chemicals produced by the cells themselves or in their environment, and is a significant mechanism of directional migration of cells. A well-known chemotaxis model was proposed by Keller and Segel ([24,25]) in the 1970s, which describes the aggregation processes of the cellular slime mold Dictyostelium discoideum. Since then, the following quasi-chemotaxis-only model and its variations have been widely studied by many authors, where the main issue of the investigation was whether the solutions to the models are bounded or blow-up (see e.g., Herrero and Velázquez [13], Nagai et al. [33], Winkler et al. [55,57], the survey [3]).
One important extension of the classical Keller-Segel model to a more complex cell migration mechanism was proposed by Chaplain and Lolas ( [7,8]) in order to describe processes of cancer invasion. In fact, let u = u(x, t) denote the density of the tumour cell population, v = v(x, t) represent the concentration of a matrix-degrading enzyme (MDE) and w = w(x, t) stand for the density of the surrounding tissue (extracellular matrix (ECM)).
In realistic situations, the renewal of the ECM occurs at much smaller timescales than its degradation (see [43,35,22,30,32,48,51]). Therefore, a choice of η = 0 on (1.2) seems justified (see [43,35,22,30,32,48,51]). The models mentioned above described the random part of the motion of cancer cells by linear diffusion, however, from a physical point of view migration of the cancer cells through the ECM should rather be regarded like movement in a porous medium, and so we are led to considering the cell motility D a nonlinear function of the cancer cell density. Inspired by the analysis, in this paper,we consider the following chemotaxis-haptotaxis system with nonlinear diffusion (see also [44,3,52]) u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), w(x, 0) = w 0 (x), x ∈ Ω (1. 3) in a bounded domain Ω ⊂ R N (N ≥ 1) with smooth boundary ∂Ω. The origin of the system was proposed by Chaplain and Lolas ( [7,8]) to describe cancer cell invasion into surrounding healthy tissue. Here we assume that D(u) is a nonlinear function and satisfies D ∈ C 2 ([0, ∞)) and D(u) ≥ C D u m−1 for all u > 0 (1. 4) with some C D > 0 and m > 0. Moreover, if, D(u) fulfills D(u) > 0 for all u ≥ 0, (1.5) so the diffusion is nondegenerate and the solutions may be considered in the sense of classical.
Throughout this paper, the initial data (u 0 , v 0 , w 0 ) are assumed that for some ϑ ∈ (0, 1) ( (1.7) However, they leave a question here: "whether the global solutions are bounded". If N ≥ 2, the global boundedness of solutions to (1.3) has been constructed for m > 2− 2 N (see [27,52]) with the help of the boundness of ∇v L l (Ω×(0,T )) (1 ≤ l < N N −1 )). Recently, we ( [59]) extended these results to the cases m > 2N N +2 by using the boundness of ∇v L 2 (Ω×(0,T )) . More recently, if µ χ is large enough, Jin [23] (see also [20]) proved that system (1.3) admits a global bounded solution for any m > 0. However, we should point that the cases 0 < m ≤ 2N N +2 and small µ χ remain unknown even in the case for the chemotaxis-only system (1.3), that is, w ≡ 0 in system (1.3). In this paper, we firstly use the boundedness of t (t−1) + Ω u γ 0 +1 (see Lemma 3.6) for some γ 0 > 1, which is a new result even for chemotaxis-only system (1.3). Then, applying the standard testing procedures, we can derive the uniform boundedness of ∇v in L l 0 (Ω) for some l 0 > 2. We emphasize that the spontaneous boundedness information on ∇v in L l 0 (Ω) (see (3.40)) plays a key role in this process. Using the L l 0 -boundedness of ∇v and L 1 -boundedness of u, we can then acquire the uniform bounds of u in arbitrary large L p (Ω) provided that the further restriction on m is satisfied (see the proof of Lemmas 3.10-3.14). Finally, combining with Moser iteration method and L p -L q estimates for Neumann heat semigroup, we finally established the L ∞ bound of u (see . Motivated by the above works, this paper will focus on studying the relationship between the exponent m and the global existence of solutions to chemotaxis-haptotaxis model (1.3) with nonlinear diffusion. In fact, the aim of the present paper is to study the quasilinear in the classical sense. Moreover, both u, v and w are bounded in Ω × (0, ∞), that is, there exists a positive constant C such that then γ * * = (µ * +1)(N +µ * −1)  [31]). Here the assumption m > 2N N +2 (see [59]) or m > 2 − 2 N (see [27,31,52]) are intrinsically required. (ii) Obviously, for any N ≥ 1, (iii) In the case N = 2, by using µ > 0, then 8 4+(µ * +1) 2 < 1, our result improves the result of [41] and [64], in which the assumption m = 1 or m > 1 are intrinsically required.
(v) The chemotaxis-haptotaxis system therefore has bounded solutions under the same condition on m as the pure chemotaxis system with w ≡ 0 without logistic source (see [45]).
In the case of possibly degenerate diffusion, system (1.3) admits at least one global bounded weak solution: The rest of this paper is organized as follows. In the following section, we recall some preliminary results. Section 3 is devoted to a series of a priori estimates and then prove Theorem 1.1. In Section 4, applying the existence of classical solutions in the non-degenerate case, we will then complete the proof of theorem 1.2 by an approximation procedure in Section 3.

Preliminaries and main results
Before proving our main results, we will give some preliminary lemmas, which play a crucial role in the following proofs. As for the proofs of these lemmas, here we will not repeat them again.
According to the above existence theory, for any s ∈ (0, T max ), (u(·, s), v(·, s), w(·, s)) ∈ C 2 (Ω). Without loss of generality, we can assume that there exists a positive constant K such that

A priori estimates
The main task of this section is to establish for estimates for the solutions (u, v, w) of problem (1.3). To this end, in straightforward fashion one can check the following boundedness for u, which is common in chemotaxis (or chemotaxis-haptotaxis) with logistic source (see e.g. [59,54,52,27]).
Lemma 3.1. There exists C > 0 such that the solution of (1.3) satisfies . Since, the third component of (1.3) can be expressed explicitly in terms of v. This leads to the following a one-sided pointwise estimate for −∆w (see e.g. [49,41,44]): Now we proceed to establish the main step towards our boundedness proof. To this end, let us collect some basic estimates for u and v in comparatively large function spaces. In fact, relying on a standard testing procedure, we derive the following Lemma:
Proof. For any k > 1, we integrate the left hand of (3.4) and use Lemma 3.2 then get where κ is the same as (3.3). This directly entails (3.4).
Due to the presence of logistic source, some useful estimates for u can be derived.
Lemma 3.4. (see [52,27,59]) Assume that µ > 0. There exists a positive constant K 0 such and t+τ t Ω where we have set In order to establish some estimates for solution (u, v, w), we first recall the following lemma proved in [52] (see also [27,59]).
Proof. Multiplying (1.3) 1 (the first equation of (1.3)) by u k−1 and integrating over Ω, we get according to the nonnegativity of w. Integrating by parts to the first term on the right hand side of (3.10) and using the Young inequality, we obtain (3.11) On the other hand, due to Lemma 3.3, we have Furthermore, inserting (3.11)-(3.12) into (3.10), we conclude that for all t ∈ (0, T max ), We proceed to estimate both integrals on the right of (3.9) in a straightforward manner. (3.14) If µ > 0, then for all 1 < γ 0 < µ * , there exists a positive constant C which depends on γ 0 such that Proof. Multiplying (1.3) 1 by u k−1 , integrating over Ω and using w ≥ 0, we get We now estimate the right hand side of (3.17) terms by terms. To this end, integrating by parts to the first term on the right hand side of (3.17), we obtain for any ε 1 > 0, where where and κ is give by (3.3).
On the other hand, in view of k > 1, we also derive that For any t ∈ (0, T max ), applying the Gronwall Lemma to the above inequality shows that Next, a use of Lemma 2.3 and (2.2) leads to and for all t ∈ (0, T max ), where λ 0 is the same as Lemma 2.3. On the other hand, choosing , with the help of (3.19) and (3.21), a simple calculation shows that Thus, by using the Young inequality, we derive that there exists a positive constant C 4 such Thereupon, combining with the arbitrariness of ε and the Hölder inequality, (3.15) and (3.16) holds. The proof of Lemma 3.6 is completed. When by making use of above lemma, we can derive the following results on the bound u for in an L k space for any k > 1.
then for all k > 1, there exists a positive constant C which depends on k such that Proof. This directly results from Lemma 3.6 and the fact that In the following, we always assume that since, case µ ≥ max s≥1 λ 1 s+1 0 (χ + ξ w 0 L ∞ (Ω) ) has been proved by Corollary 3.1.
We proceed to establish the main step towards our boundedness proof. The following lemma can be used to improve our knowledge on integrability of ∇v, provided that µ > 0.
Its repeated application will form the core of our regularity proof.
Proof. Let γ 0 and µ * be same as Lemma 3.6. For the above 1 < γ 0 < µ * , we choose β = γ 0 +1 2 in (3.31). Then by using the Young inequality, we derive that for some positive constant C 1 , Here κ 0 is the same as (3.31). Inserting (3.41) into (3.31), we conclude that there exists a positive constant C 2 such that which combined with (3.16) implies that by an ODE comparison argument. On the other hand, for any β > 1, it then follows from Lemma 2.2 that there exist positive constants κ 1 and κ 2 such that ∇v ) ≤ κ 2 ( ∇|∇v| β 2 L 2 (Ω) + 1) where κ 2 is the same as (3.44). Therefore, collecting (3.44) and (3.45), we have therefore, in view of (3.16), by using an ODE comparison argument again, we have with some positive constant C 6 , which yields (3.40), and hence completes the proof.

48)
where C is a positive constant.
Proof. Collecting Lemma 3.5 and Lemma 3.8, we can derive (3.48) by using the Young inequality.
We next plan to estimate the right-hand sides in the above inequalities appropriately by using a priori information provided by Lemma 3.8 and Lemma 3.1. Here the following lemma will will play an important role in making efficient use of the known L (γ 0 +1)(N+γ 0 −1) N (Ω) bound for ∇v. The following lemma provides some elementary material that will be essential to our bootstrap procedure.
Proof. Due to According to the estimate of ∇v in (3.74) along with the Hölder inequality we derive that there exists a positive constant C 3 > 0 such that .

(3.77)
In view of k > max{|1 − m| + q 0, * p 0, * −1 An application of the Gagliardo-Nirenberg inequality (see Lemma 2.2) implies that for some positive constants C 4 as well as C 5 such that so that, applying the Young inequality implies that which together with the Young inequality again yields to and some positive constants C 8 and C 9 . Now, in view of 2p 0 > N (see (3.75)), then by Sobolev imbedding theorems, we derive from (3.74 so that, combined with (3.80) implies that whereas a standard ODE comparison argument shows that (3.70) holds.
with some positive constants C 7 as well as C 8 and On the other hand, again, it infers by the Gagliardo-Nirenberg inequality (Lemma 2.2) that there are C 9 > 0 and C 10 > 0 ensuring Inserting (3.91)-(3.92) into (3.90) and using k > and k > 2 − 2 N − m and Lemma 2.2, we derive that for the above δ > 0, (3.93) Finally, with the help of (3.1) and by the Sobolev inequality, the Young inequality and (3.85), we conclude that for the above δ > 0, there exist positive constants C 13 , C 14 as well as C 15 and C 16 such that and some positive constant C 16 . Therefore, letting y := Thus a standard ODE comparison argument implies boundedness of y(t) for all t ∈ (0, T max ).
Employing Lemmas 3.12-3.13, we can prove the following lemma.
Let Ω ⊂ R N (N ≥ 1) be a bounded domain with smooth boundary. Furthermore, assume that m > 2N N +γ * with N ≥ 1, where γ * is given by (3.56). Then for any k > 1, there exists a positive constant C such that u(·, t) L k (Ω) ≤ C for all t ∈ (0, T max ). (3.96) Along with the Duhamel's principle and L p -L q estimates for Neumann heat semigroup, the above lemma yields the following Lemma. where (e t∆ ) t≥0 is the Neumann heat semigroup in Ω. Using Lemma 3.14, we follow the L p -L q estimates for Neumann heat semigroup to obtain for any t ∈ (0, T max ), we obtain v(·, t) W 1,∞ (Ω) This lemma is proved.
Applying Lemma 3.14 and Lemma 3.15, a straightforward adaptation of the well-established Moser-type iteration procedure [1] allows us to formulate a general condition which is sufficient for the boundedness of u. Proof. Firstly, by Lemma 3.14, we obtain that for any k > 0, u(·, t) L k (Ω) ≤ α k for all t ∈ (0, T max ), (3.101) where α k depends on k. Multiplying the first equation of (1.3) by ku k−1 with k ≥ max{2m, N+ 2}, integrating it over Ω, then using the boundary condition ∂u ∂ν = 0 and combining with Lemma 3.15, we obtain In what follows, we estimate the last two terms of (3.102).
When m ≥ 1, then, so that, by (3.102), we have (3.103) In order to take full advantage of the dissipated quantities appearing on the left-hand side herein, for any δ > 0, we first invoke the Gagliardo-Nirenberg inequality which provides C 3 > 0, C 4 > 0 as well as C 5 > 0 and C 6 > 0 such that .

(3.106)
Notice that we further obtain .

(3.107)
In what follows, we use Moser iteration method to show the L ∞ estimate of u. Take k i = and thus From this, it follows that (3.100) is valid with some positive constant. When 0 < m < 1, noticing that u k (u + ε) 1−m ≤ u k+1−m + ε 1−m u k , then by (3.101) and ε ∈ (0, 1), we derive from Lemma 3.15 that and .
Substituting the above two inequalities into (3.102), we obtain .
By virtue of (2.1) and Lemmas 3.15-3.16, we are now in a position to prove Theorem 1.1.
Proof of Theorem 1.1.
Therefore, for any ε ∈ (0, 1), the regularized problem of (1.3) is presented as follows  We proceed to establish the main step towards the boundedness of weak solutions to (1.3).
To this end, firstly, from (2.1) and Lemmas 3.15-3.16, we can easily derive the following estimates for u ε and v ε , which plays an important role in proving Theorem 1.2.
To achieve the convergence result, we need to derive some regularity properties of time derivatives.
Next we shall prove that (u, v, w) is a weak solution of problem (1.3). To this end, multiplying the first equation as well as the second equation and third equation in (4.1) by ϕ ∈ C ∞ 0 (Ω × [0, ∞)), we obtain  We can now easily prove our main result.
The proof of Theorem 1.2 A combination of Lemma 4.1 and Lemma 4.4 directly leads to our desired result.