BOUNDEDNESS AND GLOBAL SOLVABILITY TO A CHEMOTAXIS-HAPTOTAXIS MODEL WITH SLOW AND FAST DIFFUSION

. In this paper, we deal with the following coupled chemotaxis-haptotaxis system modeling cancer invasion with nonlinear diﬀusion, + , where Ω ⊂ R N ( N ≥ 3 ) is a bounded domain. Under zero-ﬂux boundary conditions, we showed that for any m > 0, the problem admits a global bounded weak solution for any large initial datum if χµ is appropriately small. The slow diﬀusion case ( m > 1) of this problem have been studied by many authors [14, 7, 19, 23], in which, the boundedness and the global in time solution are established for m > 2 NN +2 , but the cases m ≤ 2 NN +2


School of Mathematical Sciences, South China Normal University
Guangzhou, 510631, China

(Communicated by Michael Winkler)
Abstract. In this paper, we deal with the following coupled chemotaxishaptotaxis system modeling cancer invasion with nonlinear diffusion, where Ω ⊂ R N (N ≥ 3 ) is a bounded domain. Under zero-flux boundary conditions, we showed that for any m > 0, the problem admits a global bounded weak solution for any large initial datum if χ µ is appropriately small. The slow diffusion case (m > 1) of this problem have been studied by many authors [14,7,19,23], in which, the boundedness and the global in time solution are established for m > 2N N +2 , but the cases m ≤ 2N N +2 remain open.
1. Introduction. In this paper, we consider the following coupled chemotaxishaptotaxis model where m > 0, Q = Ω × R + , Ω ⊂ R N (N ≥ 3) is a bounded domain with smooth boundary. u, v, ω represent the cancer cell density, urokinase Plasminogen Activator (uPA) protease concentration, and the extracellular matrix (ECM) density respectively, χ, ξ ≥ 0 are the chemotactic and haptotactic coefficients respectively, µu(1 − u − ω) with µ > 0 is the proliferation or death of cancer cell according to a generalized logistic law including competition for space with the ECM, −v +u is the we refer to [3,4,11,18] for the study of global solutions and blow up solutions. However, in addition to migration, the cell density should also be effected by the proliferation or death of cells, so a logistic damping term µu(1 − u) characterizing the death and proliferation of cells is introduced into this model. It has been detected that the logistic damping of cancer cell densities will weaken the aggregation behavior, and prevent blow up. In fact, in the two dimensional space, it has been shown that the solutions will always exist globally [17]; and in the three dimensional case, the solutions will exist globally for large µ [17,22]. This chemotaxis-haptotaxis model was first introduced by Chaplain and Lolas [2], it described the process of cancer cell invade the surrounding normal tissue. In this model, the random diffusion of cancer cells is characterized by linear isotropic diffusion, that is the model (1) with m = 1. For this case, the global classical solution is obtained in the two dimensional space by [13], and for the three dimensional case, the global classical solution is obtained only for large µ χ [1], and remains open for small µ χ . However, from a physical point of view, the equation modelling the migration of cancer cells should rather be regarded as nonlinear diffusion [10,14], so some researchers are led to consider the nonlinear diffusive model, in which, the cell mobility is described by a nonlinear function of the cancer cell density, and the most common form is the porous medium diffusion. In 2011, Tao and Winkler [14] considered the nonlinear diffusion case without degenerate, that is the model (1) with ∆u m replaced by ∆((u + ε) m−1 u), and the global in time classical solutions are established for the following cases where N is the space dimension. However, they leave a question here: "whether the global solutions are bounded". Recently, Li, Lankeit [7] and Wang [19], showed the global solvability of classical solution with ε > 0 (or weak solution with ε = 0) for any m > 2N −2 N , in particular, the boundedness of the solutions also are established. Recently, Zheng [23] extended these results to the cases m > 2N N +2 . But the cases 1 < m ≤ 2N N +2 remain unknown. On the other hands, for the fast diffusion cases, that is 0 < m < 1, as far as we know, there are no relevant researches.
In the present paper, we consider this problem in N dimensional space with N ≥ 3, and we extend the above results to any m > 0. Firstly, we consider the regularized problem (10), and showed the boundedness and global solvability of classical solutions for any m > 0 under the assumption that χ µ is appropriately small. Then, by letting ε → 0, we further obtain the boundedness and global solvability of weak solutions.
Remark 1. For simplicity, in what follows, we always assume that m < 2, since that the case m > 2N N +2 have been completely solved.

2.
Preliminaries. We first give some notations, which will be used throughout this paper.
Next, we give the definition of weak solutions.
Before going further, we list some lemmas, which will be used throughout this paper. Firstly, we give the Gagliardo-Nirenberg interpolation inequality [8] as follows and s > 0 is arbitrary.
We list the following lemma [12].
is absolutely continuous such that Then We also give a generalized lemma of Lemma 2.3.
is absolutely continuous, and satisfies where Proof. Noticing that f ≤ f 1+σ + 1, then By Lemma 2.3, we obtain By (6), we further have (7) is obtained. By [5], we give the following Lemma.
where τ > is a fixed constant. Then the following problem admits a unique solution u with u ∈ L p loc ((0, +∞); W 2,p (Ω)) and u t ∈ L p loc ((0, +∞); where M is a constant independent of τ .
In what follows, all these constants C,C, M , C i , M i denote some different constants, which are independent of ε, τ and T max . If no special explanation, they depend at most on m, N , µ, ξ, χ, u 0 , v 0 , ω 0 and Ω. In fact, It is easy to see that τ = 1 if T max ≥ 2. It is worth mentioning that if τ < 1, meaning T max < 2, and all these estimates of this form t t−τ · · · ds in this paper can be replaced by Tmax 0 · · · ds. So all these constants are independent of τ . sup sup where C 1 , C 2 are independent of T max , τ and ε.
Proof. By a direct calculation, it is easy to see that (16) holds. Integrating the first equation of (10) directly, we obtain Then by lemma 2.4 we obtain (17). Recalling the second equation of (10), using (17) and combining with Lemma 2.5, we obtain (18). The proof is complete.
where C depends on µ.
where M 1 is independent of µ, M 2 depends on µ, and both of them depend on r.

Summing up, we have
).
which implies (20), and the proof is complete.
Using this lemma, we can further prove that Lemma 3.5. Assume (B). Then if χ µ is sufficiently small, we have sup sup where C is independent of T max and ε.
Proof. By Lemma 3.4, it is easy to see that if χ µ is sufficiently small, then for any r > 0. By Duhamel's principle, we see that the solution v ε can be expressed as follows where {e t∆ } t≥0 is the Neumann heat semigroup in Ω, for more details of Neumann heat semigroup, please refer to [20]. By (25) with r = N , we obtain for any Multiplying the first equation of (10) by u r−1 ε with r > max{2N, 4m}, then integrating it over Ω, and combining with (23), we obtain Here C is independent of r. In what follows, we estimate the last two terms of (26). When m ≥ 1, by (26), we further have By Gagliardo-Nirenberg interpolation inequality, it follows and Substituting the above two inequalities into (27), we obtain where C is independent of r.
By (19) and (24), we obtain It means (15) holds. By the expression of ω ε and (11), and we note that then for any T > 0, where C T depends on T and is independent of ε.
Letting ε → 0 + , we also obtain Noticing that ∇u m ∈ L 2 (Q T ), then we also have