On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion

This paper deals with a parabolic-parabolic-ODE chemotaxis 
haptotaxis system with nonlinear diffusion 
\begin{eqnarray*}\label{1a} 
\left\{ 
\begin{split}{} 
&u_{t}=\nabla\cdot(\varphi(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, \\ 
&w_{t}=-vw, 
\end{split} 
\right. 
\end{eqnarray*} 
under Neumann boundary conditions in a smooth bounded domain 
$\Omega\subset \mathbb{R}^{2}$, where $\chi$, $\xi$ and $\mu$ are 
positive parameters and $\varphi(u)$ is a nonlinear diffusion 
function. Firstly, under the case of non-degenerate diffusion, it is 
proved that the corresponding initial boundary value problem 
possesses a unique global classical solution that is uniformly 
bounded in $\Omega\times(0,\infty)$. Moreover, under the case of 
degenerate diffusion, we prove that the corresponding problem admits 
at least one nonnegative global bounded-in-time weak solution. 
Finally, under some additional conditions, we derive the temporal 
decay estimate of $w$.

1. Introduction. In this paper, we consider the following chemotaxis-haptotaixs system with nonlinear diffusion x ∈ Ω, t > 0,

PAN ZHENG, CHUNLAI MU AND XIAOJUN SONG
where Ω ⊂ R 2 is a bounded domain with smooth boundary ∂Ω, ∂ ∂ν denotes the differentiation with respect to the outward normal derivative on ∂Ω, and ϕ ∈ C 2 ([0, ∞)) is the nonlinear diffusion function. The parameters χ, ξ and µ are positive and the initial data (u 0 , v 0 , w 0 ) is supposed to be satisfied the following conditions w 0 ∈ C 2+α (Ω) with α ∈ (0, 1) w 0 > 0 in Ω and ∂w 0 ∂ν = 0 on ∂Ω. (2) The oriented movement of biological cells or organisms in response to a chemical gradient is called chemotaxis. The pioneering works of chemotaxis model were introduced by Patlak [22] in 1953 and Keller and Segel [18] in 1970, and we refer the reader to the survey [11,13,14] where a comprehensive information of further examples illustrating the outstanding biological relevance of chemotaxis can be found. In recent years, chemotactic mechanisms have also been detected to be crucial in the process of cancer invasion, where they usually interact with haptotaxis, the correspondingly directed cell movement in response to gradients of non-diffusible signals. The combination of these two cell migration mechanisms was initially proposed by Chaplain and Lolas in [4,5] to describe cancer cell invasion into surrounding healthy tissue. More precisely, their model accounts for both chemotactic migration of cancer cells towards a diffusible matrix-degrading enzyme (MDE) secreted by themselves, and haptotactic migration towards a static tissue, also referred to as extracellular matrix (ECM). In this context, u(x, t) represents the density of cancer cell, v(x, t) denotes the concentration of MDE, and w(x, t) stands for the density of ECM. In addition to random movement, cancer cells are supposed to bias their movement both towards increasing concentrations of urokinase plasminogen activator by chemotaxis (see [3]), and towards increasing densities of the non-diffusible ECM through detecting the macromolecules adhered therein by haptotaxis (see [2]). It is assumed that the cancer cells undergo birth and death in a logistic manner, competing for space with the ECM. The MDE is assumed to be produced by cancer cells, and to diffuse and decay, whereas the ECM is stiff in the sense that it does not diffuse, but it could be degraded upon contact with MDE.
In order to better understand model (1), let us mention some previous contributions in this direction. In recent years, the following initial boundary value problems have been studied by many authors x ∈ Ω, t > 0, where τ ∈ {0, 1}, χ > 0, ξ > 0, µ > 0 and Ω ⊂ R n (n ≥ 1) is a bounded domain with smooth boundary ∂Ω. When τ = 0, i.e. the diffusion rate of the MDE is much greater than that of cancer cells [5]. Moreover, similar quasi-steady-approximations for corresponding chemoattractant equations were frequently used to study classical chemotaxis systems (for instance [17,23]). For the special case ϕ(u) = 1 in (3), Tao and Wang [27] proved that model (3) possesses a unique global bounded classical solution for any µ > 0 in two space dimensions, and for large µ > 0 in three space dimensions. Recently, in [31], Tao and Winkler studied global boundedness for model (3) with ϕ(u) = 1 under the condition µ > (n−2)+ n χ, moreover, in additional explicit smallness on w 0 , they gave the exponential decay of w in the large time limit.
When τ = 1, for the special case ϕ(u) = 1 in (3), Tao and Wang [26] proved that model (3) possesses a unique global-in-time classical solution for any χ > 0 in one space dimension, and for small χ µ > 0 in two and three space dimensions. Later, Tao [24] improved the result of [26] for any µ > 0 in two space dimension.
Tao and Winkler [29] proved that model (3) possesses at least one nonnegative global classical solution, however, their boundedness is left as an open problem. In [12], Hillen et.al studied the global boundedness and asymptotic behavior of model (3) with ϕ(u) = 1 in one space dimension and proposed an open problem about boundedness in the higher-dimensional case. Recently, for the case ϕ(u) = 1 in (3), Tao [25] showed that under appropriate regularity assumption on the initial data (u 0 , v 0 , w 0 ), the corresponding initial-boundary problem possesses a unique classical solution which is global in time and bounded in two space dimensions. To the best of our knowledge, there exist few boundedness results addressing the fully parabolic-parabolic-ODE chemotaxis-haptotaxis model in the higher-dimensional version.
In the present paper, our main purpose is to give a positive answer about the question of global boundedness in [12,29]. Motivated by the above works, we consider global boundedness and temporal decay estimate of w for model (1) under some suitable conditions. The crucial assumption in our result is related to the diffusion function ϕ(u). Here, we suppose that the function ϕ satisfies and with some c 0 > 0 and m > 1. If in addition to (5) and (6), ϕ(s) satisfies ϕ(s) > 0 for all s ≥ 0, then the first equation in (1) becomes uniformly parabolic, thus the solution of (1) may be considered in the classical sense.
Our main results in this paper are stated as follows. Firstly, we consider global boundedness of model (1) in the case of non-degenerate diffusion.
In the absence of (7), the first equation of (1) may be degenerate at u = 0, thus in general there is no classical solution. Hence, we shall study global weak solutions in (1). Before stating our second result, we first give the definition of weak solutions of problem (1).
Finally, under some addition conditions, we consider decay of w.
Then there exist positive constants κ and C such that the third solution component w satisfies the following decay estimate This paper is organized as follows. In Section 2, we shall state some preliminary results which are important for our main proofs. In Section 3, we consider global boundedness of classical solutions for model (1) under some suitable conditions and prove Theorem 1.1. In Section 4, we concern with global weak solutions of problem (1) and prove Theorem 1.3. In Section 5, we give the decay estimate of w under some additional assumptions and prove Theorem 1.4.

2.
Preliminaries. We first state one result concerning local-in-time existence of a classical solution to model (1).
Lemma 2.1. Let χ > 0, ξ > 0 and µ > 0, and assume that the function ϕ satisfies (5)- (7). Then for any initial data (u 0 , v 0 , w 0 ) fulfilling (2), there exists a maximal existence time T max ∈ (0, ∞] such that model (1) possesses a unique classical solution Finally, if T max < +∞, then Proof. The local-in-time existence of classical solution to model (1) is well-established by a fixed point theorem in the context of chemotaxis-haptotaxis systems. By the maximum principle, it is easy to obtain that u ≥ 0 and v ≥ 0 for all (x, t) ∈ Ω × [0, T max ). Integrating the third equation in (1), it follows from (2) and The proof is quite standard, for details, we refer the readers to [6,29,32,37,15,34].
Next, we give the basic property on mass for u.
is a solution for model (1). Then there exists a constant m * > 0 such that the first component u of the solution to (1) satisfies the following estimate Proof. Integrating the first equation in (1), we deduce from w > 0 that According to Hölder's inequality, we have By the comparison argument of ODE, we derive The proof of Lemma 2.2 is complete.
Moreover, we show the following one-sided pointwise estimate for −∆w, which will be served as a cornerstone for our subsequent analysis.
Finally, let us collect some basic statements about the Gagliardo-Nirenberg inequality which will be used in the forthcoming proof of L p -boundedness of solutions for model (1). For details, we refer the readers to [7,21,37] (see also [28]).

Lemma 2.4.
Let Ω ⊂ R n be a bounded domain with smooth boundary and assume that r ∈ (0, p) and φ ∈ W 1,2 (Ω) ∩ L r (Ω). Then there exists a positive constant C GN such that holds with λ ∈ (0, 1) satisfying 3. Global boundedness. In this section, inspired by Liu and Tao in [20] (see also [25]), we first establish an energy-type estimates for Ω u ln udx and Ω |∇v| 2 dx. Next, we further establish a bound for Ω u 2 dx and Ω |∇v| 4 dx via coupled estimate techniques. Thirdly, we build a bound for Ω u p dx for any p > 1. Finally, we derive a L ∞ (Ω)-estimate for u and prove Theorem 1.1. To do this, we need to obtain the following a priori estimates for the solutions of model (1). Firstly, in the two-dimensional case, by using the properties of the Neumann heat semigroup ( [36]) and Lemma 2.2, we give a L p -estimate on v.
Proof. The main idea of the proof comes from that in [20,25]. For convenience of the readers, we give the details. Multiplying the first equation in (1) by (1 + ln u) and integrating by parts, we have for all t ∈ (0, T max ). With the aid of Lemma 2.2, Lemma 2.3 and Lemma 3.1, it follows from the same way as in [25], we can obtain · K(2) + (ξM + µ)m * + µ e ||w 0 || L ∞ (Ω) |Ω|. In order to cancel the first term on the right of (27), we again use the same way as in [25] to derive Combining (27) with (28), we have d dt Let it follows from (29) and Lemma 3.1 in [25] that where C 3 := L|Ω| + C 2 . By the comparison argument of ODE, we derive which implies that (24) and (25) hold.
Proof. Multiplying the first equation in (1) by u and integrating by parts, we have According to (5)- (7), without loss of generality, assume that ϕ(u) ≥ c 0 (u + 1) m−1 , thus we can obtain By Young's inequality, we derive from m > 1 that It follows from Lemma 2.3 that Due to the fact that w > 0, it is easy to see that Combining (34)-(38), we have By using Young's inequality and Lemma 3.1, we derive where . By using Young's inequality again, we obtain where C 5 := 16 27µ 2 (ξM + 2µ) 3 |Ω|. Thus, it follows from (39)-(41) that where C 6 := C 4 + C 5 . Adding Ω u 2 dx to both sides of (42) and using the inequality thanks to the Young inequality, we have where C 7 := C 6 + 4 27µ 2 |Ω|. In order to deal with the first integral term on the right of (44), we need to establish an energy inequality for Ω |∇v| 4 dx.
for all t ∈ (0, T max ). Next, we shall show that the three integrals on the right of (46) can be controlled by 4c0 (m+1) 2 Ω |∇(u + 1) m+1 2 | 2 dx + Ω |∇|∇v| 2 | 2 dx. One main contribution of a recent work [16] is to deal with the boundary-related integrals, thus for any ε 1 > 0, there exists C 8 (ε 1 ) > 0 such that By using Young's inequality with ε 2 > 0, we obtain It follows from the Gagliardo-Nirenberg inequality in Lemma 2.4 and (25) that where C 9 , C 10 , C 11 are positive constants and we have used the fact that the space dimension is two. By using the Gagliardo-Nirenberg inequality again, we infer from m > 1, Lemma 2.2 and Young's inequality that where C 12 > 0, C 13 > 0 and C 14 (ε 2 ) > 0.
Proof. Multiplying the first equation in (1) by u p−1 , (p > 2m − 1) and integrating by parts, we have According to (5)- (7), we obtain By Young's inequality, we derive It follows from Lemma 2.3 that Due to the fact that w > 0, it is easy to see that Combining (55)-(59), we have According to Lemma 3.1 and Lemma 3.3, there exist C 16 > 0 and K 1 (q) such that and Ω |∇v| q dx ≤ K 1 (q) for all q ∈ [4, ∞) and t ∈ (0, T max ).
Lemma 3.5. Let (u, v, w) be a solution of model (1) and assume that the conditions in Theorem 1.1 hold. Then there exists C > 0 such that Proof. By Lemma 3.4 and using parabolic regularity theory (see Lemma 4.1 in [15] or Lemma 1 in [19]) to the second equation in (1), there exists C 21 > 0 such that Combining (60) with (70), we deduce from Young's inequality that with positive constants C 22 and C 23 . In view of Lemma 3.4 and using the standard Moser-Alikakos iteration ( [1,30])(see also Lemma 4.2 in [35]), we derive that u is uniformly bounded in Ω × (0, T max ). The proof of Lemma 3.5 is complete. Now we begin with the proof of Theorem 1.1.
Proof of Theorem 1.1. By Lemma 3.5, it is easy to see that there exists a positive constant C 24 > 0 such that ||u(·, t)|| L ∞ (Ω) ≤ C 24 for all t ∈ (0, T max ).
With the aid of the blow up criterion (14) in Lemma 2.1, we know T max = ∞. By combining (13) with (63), we can obtain the desired boundedness result. The proof of Theorem 1.1 is complete.

Global weak solutions.
In general, the degenerate diffusion case of (1) might not have classical solutions, thus in order to justify all the formal arguments, we need to introduce the following approximating equation of (1): where ε ∈ (0, 1), ϕ ε (s) = ϕ(s + ε) for all s ≥ 0, and the initial data w 0ε w 0 weakly star in L ∞ (Ω).
Now in conjunction with (74), (76), (77) and the Aubin-Lions lemma (Theorem III.2.3 in [33]), there exists a subsequence ε = ε j 0 as j → ∞ such that w ε w weakly in star L ∞ (Ω × (0, T )) and ∇w ε ∇w weakly in star L ∞ (Ω × (0, T )). (78) Multiplying the first, second and third equations of (72) by ζ, η and θ, respectively, and then integrating by party, we see that (u ε , v ε , w ε ) satisfies and as well as By using (73) and (78) in passing to the limit in each term of the identities (79)-(81), we obtain and as well as Hence, it is easy to see that (u, v, w) is a global weak solution for (1). Finally, the boundedness statement is a straight forward consequence of the proof of Theorem 1.1. The proof of Theorem 1.3 is complete.

5.
Decay of w. In this section, inspired by Hillen et al. in [12], we consider the decay estimate of w under a suitable smallness condition on w 0 and prove Theorem 1.4. The proof is mainly based on a lower bound for the mass m(t) = Ω u(x, t)dx.
To do this, we need the following lemmas.
Proof. The proof is mainly based on the arguments in [12]. According to the strong maximum principle and the hypothesis u 0 ≡ 0 in (2), then we know that u is positive in Ω × (0, ∞). Thus, we may multiply the first equation in (1) by u γ−1 , γ ∈ (0, 1) and integrate by parts over Ω to obtain According to the conditions (5)- (7), we obtain By Young's inequality, we deduce from (70), Hölder's inequality and 1 < m < 2 that It follows from Lemma 2.3 and (63) that By the Gagliardo-Nirenberg inequality in Lemma 2.4, we deduce from Lemma 2.2 that there exist C 25 > 0 and C 26 > 0 such that where we have used the fact that the space dimension is two, and Due to the fact that m > 1, it is easy to see that Thus, using Young's inequality, it follows from (90) and (92) that there exists a positive constant C 27 such that Finally, we denote η := µ − µ||w 0 || L ∞ (Ω) > 0, due to the condition ||w 0 || L ∞ (Ω) < 1. By using the fact that w t ≤ 0, we have Hence, collecting (86)-(89), (93) and (94), we derive Now, picking γ ∈ (0, 1) sufficiently close to one such that and γ − m + 1 γ ∈ (0, 1), due to the condition 1 < m < 2. Therefore, we infer from Young's inequality that where C 28 := Since γ ∈ (0, 1), it follows from Hölder's inequality that Thus, we deduce from (99) that The proof of Lemma 5.1 is complete.
Next, we shall give the following lower bound estimate for v. For the details of the proof, please see Corollary 3.3 in [12] (see also Lemma 4.5 in [31] or Lemma 2.1 in [8,9,10]).