Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion

This paper deals with a parabolic-elliptic-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), \\ 
&0=\Delta v-v+u, \\ 
&w_{t}=-vw, 
\end{split} 
\right. 
\end{eqnarray*} 
under Neumann boundary conditions in a smooth bounded domain 
$\Omega\subset \mathbb{R}^{n}$ $(n\geq1)$, where $\chi$, $\xi$ and 
$\mu$ are positive parameters and $\varphi(u)$ is a nonlinear 
diffusion. Under the non-degenerate diffusion and some 
suitable assumptions on positive parameters $\chi,\xi,\mu$, it is 
shown 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 
degenerate diffusion, it is proved that the corresponding problem 
admits at least one nonnegative global bounded-in-time weak 
solution. Finally, for the suitably small initial data $w_{0}$, we give the decay estimate of $w$.

1. Introduction. In this paper, we consider the following chemotaxis-haptotaxis system with nonlinear diffusion x ∈ Ω, t > 0, where Ω ⊂ R n (n ≥ 1) is a bounded domain with smooth boundary ∂Ω, ∂ ∂ν denotes the differentiation with respect to the outward normal derivative on ∂Ω, and ϕ is the nonlinear diffusion. The parameters χ, ξ and µ are positive and the initial data (u 0 , w 0 ) is supposed to be satisfied the following conditions    u 0 ∈ C 0 (Ω) with u 0 ≥ 0 in Ω and u 0 ≡ 0, w 0 ∈ C 2+α (Ω) with α ∈ (0, 1) w 0 > 0 in Ω and ∂w 0 ∂ν = 0 on ∂Ω. ( 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 [21] in 1953 and Keller and Segel [17] 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 [6,7] 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 = 1, for the special case ϕ(u) = 1 in (3), Tao and Wang [25] 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 [23] improved the result of [25] for any µ > 0 in two space dimension. Moreover, If ϕ(u) ∈ C 2 ([0, ∞)), ϕ(0) > 0 and ϕ(u) ≥ δu m−1 , where δ > 0 and Tao and Winkler [28] proved that model (3) (3), Tao [24] 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. Moreover, Cao [4] proved the boundedness of solutions with the case ϕ(u) = 1 in (3) in three space dimensions. Furthermore, in [41], the results in [24] have been extended the nonlinear diffusion case and the decay of w was studied. Recently, the boundedness of model (3) was derived in higher-dimensional case (see [34,19]).
When τ = 0, i.e. the diffusion rate of the MDE is much greater than that of cancer cells [7]. Moreover, similar quasi-steady-approximations for corresponding chemoattractant equations were frequently used to study classical chemotaxis systems (for instance [16,22]). For the special case ϕ(u) = 1 in (3), Tao and Wang [26] proved that model (3) possesses a unique global bounded classical solution for any µ > 0 in two space dimension, and for large µ > 0 in three space dimensions. Furthermore, in [30], Tao and Winkler studied global boundedness for model (3) with ϕ(u) = 1 under the condition µ > (n−2)+ n χ, furthermore, in additional explicit smallness on w 0 , they gave the exponential decay of w in the large time limit. When w ≡ 0, model (3) is reduced to the following chemotaxis-only system This system has been widely studied by many authors in these years. In the case ϕ(u) = 1 in (5), Tello and Winkler [31] proved that the solutions of semilinear parabolic-elliptic problem are global and bounded provided that either n ≤ 2, or n ≥ 3 and µ > (n−2)χ n with χ > 0. Moreover, for any n ≥ 1 and µ > 0, the existence of global weak solution was shown under some additional conditions. Furthermore, if the logistic source f (u) ≤ a − bu k , k > 2 − 1 n , some global very weak solutions of semilinear parabolic-elliptic model were constructed by Winkler [36]. When ϕ(u) ≥ c(u + 1) p with p ∈ R and µ > 1 − 2 n(1−p)+ χ with χ > 0, Cao and Zheng [5] proved that the simplified parabolic-elliptic model (5) has a unique global classic solution, which is uniformly bounded. Recently, Wang et.al in [33] investigated the boundedness and asymptotic behavior for model (5) with the special case ϕ(u) ≥ C D u m−1 (m ≥ 1) under other additional technique conditions. In the recent paper [37], for the case of ϕ(u) = 1, f (u) = ru − µu 2 with r ≥ 0 and µ > 0, in onedimensional case, Winkler proved that going beyond carrying capacities actually is a genuinely dynamical feature of (5) provided that µ < 1 and diffusion is sufficiently weak, moreover, he investigated global boundedness and finite-time blow-up for a corresponding hyperbolic-elliptic limit problem. Furthermore, Lankeit [18] extended the results of [37] to the higher dimensional radially symmetric case. Motivated by the above works, the present paper deals with global boundedness for model (1) under some suitable conditions. The crucial assumption in our result is related to the diffusion function ϕ(u). Our main results in this paper are stated as follows.
Next, we consider weak solutions in degenerate case of (1), because in general there is no classical solution. Suppose that ϕ ∈ C 2 ([0, ∞)) satisfies ϕ(s) = c 1 s −p for all s ≥ 0, p ∈ R and some c 1 > 0. (8) Before stating our second result, we first give the definition of weak solutions of (1).
Finally, for the suitably small initial data w 0 , we consider the decay estimate of w.
Theorem 1.4. Under the same conditions of Theorem 1.1, assume that the initial data (u 0 , w 0 ) satisfies (2) and 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 show the local-in-time existence of a classical solution to model (1) and give some preliminary inequalities which are important for our proofs. In Section 3, we consider the global existence and boundedness of solutions for model (1) under some suitable conditions and prove Theorem 1.1. In Section 4, we concern with global weak solutions of (1) and prove Theorem 1.3. Finally, we prove Theorem 1.4 in Section 5.
2. Preliminaries. We first state one result concerning local-in-time existence of classical solution to model (1).

PAN ZHENG
Proof. The claims concerning local-in-time existence of classical solution to model (1) are well-established by a fixed-point argument in the context of chemotaxishaptotaxis 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 [8,28,31,38,15,33,39].
Next, based on the ideas developed in [24,30], we give the following one-sided pointwise estimate for −∆w, which will be served as a cornerstone for our subsequent analysis.
Proof. The main idea of the proof is very similar to those of Lemma 2.2 in [24] and Lemma 2.2 in [30], thus we refrain us from repeating it here.
Finally, let us collect some basic statements about the Gagliardo-Nirenberg inequality which will be used in the forthcoming proof of L γ -boundedness for model (1). For details, we refer the readers to [9,20,38] (see also [27]).

Lemma 2.3.
Let Ω ⊂ R n , n ≥ 1 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, we consider the global existence and boundedness of solutions for model (1) under some suitable conditions. To do this, we first give the following a priori estimates for model (1).
is a classical 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 ||u(·, t)|| L 1 (Ω) ≤ max ||u 0 (x)|| L 1 (Ω) , |Ω| := m * .
The proof of Lemma 3.1 is complete.
Proof. Multiplying the first equation in (1) by u γ−1 (γ > 1) and integrating by parts, we have Integrating by parts once more, it follows from the second equation in (1) and v ≥ 0 that Similarly, we deduce from Lemma 2.2 that Combining (26)-(28), we derive Due to the fact that γ < χ (χ−µ)+ , it is easy to see that By Hölder's inequality, we obtain

PAN ZHENG
Collecting (29)-(31), we have By the comparison argument of ODE, we derive for all t ∈ (0, T max ). The proof of Lemma 3.2 is complete.
Step 1. For the case p < 2 n − 1, we proceed in a similar way in [8]. By Gagliardo-Nirenberg's inequality in Lemma 2.3, it follows from Lemma 3.1 that there exist positive constants C 3 and C 4 such that where Due to the conditions p < 2 n − 1 and γ > 1, we know which implies λ ∈ (0, 1).
Hence, selecting γ > n, it follows from the Sobolev embedding theorem that where C 12 is a positive constant. In view of Lemma 3.3 and using the standard Moser-Alikakos iteration ( [1,29])(see also Lemma 4.2 in [35]), we derive that u is uniformly bounded in Ω × (0, T max ). The proof of Lemma 3.4 is complete.
Now we begin with the proof of Theorem 1.1.

PAN ZHENG
Therefore, we select ζ ∈ C ∞ 0 (Ω × [0, T )), η ∈ C ∞ 0 (Ω × [0, T )) and θ ∈ C ∞ 0 (Ω × [0, T )) for all T ∈ (0, ∞). Multiplying the first, second and third equations of (62) 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 (63) and (67) in passing to the limit in each term of the identities (68)-(70), 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, motivated by Tao and Winkler in [30], 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. Lemma 5.1. Under the same conditions of Theorem 1.1, assume that the initial data (u 0 , w 0 ) satisfies (2) and where K := min 0≤s≤||u(·,t)|| L ∞ (Ω) c 0 (s + 1) −p > 0. Then we can find β > 0 and Γ > 0 such that the first solution component u of (1) satisfies Proof. The proof is mainly based on the arguments in [30,10]. According to the 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 1 u and integrate by parts over Ω to obtain for all t > 0, where K := min 0≤s≤||u|| L ∞ (Ω) c 0 (1 + s) −p , due to the condition (6) and Theorem 1.1. The later proof is the same as Lemma 4.4 in [30], thus we refrain us from repeating it here. The proof of Lemma 5.1 is complete.
Next, we shall give the following lower bound estimate for v.
Proof. By using the strict positivity of the associated Green's function for the Helmholtz operator −∆ + 1 or alternatively by positivity properties of the Neumann heat semigroup along with a representation of the corresponding resolvent of e t∆ , it is not difficult to prove it. For details, please see Lemma 4.5 in [30] or Lemma 2.1 in [10].