BOUNDEDNESS AND GLOBAL SOLVABILITY FOR A CHEMOTAXIS-HAPTOTAXIS MODEL WITH p -LAPLACIAN DIFFUSION

. We consider a chemotaxis-haptotaxis system with p -Laplacian diffusion in three dimensional bounded domains. It is asserted that for any p > 2, under the appropriate assumptions, the chemotaxis-haptotaxis system admits a global bounded weak solution if for initial data satisﬁes certain conditions.


Introduction
In this article, we study the problem 0, u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), w(x, 0) = w 0 (x), x ∈ Ω, (1.1) where Ω is a bounded domain in R 3 with the smooth boundary, p > 2, χ, ξ ≥ 0, µ > 0, α > 0, β ≥ 0, f and g satisfy with some positive constants K i .u represent the cancer cell density, v is the matrix degrading enzyme concentration and w stands for the extracellular matrix density.A chemotaxis model was first introduced by Keller and Segel [4] in 1970, later, many modified chemotaxis models have been widely studied by many researchers.Recently, Chen and Tao [1] considered the chemotaxis-haptotaxis system They showed that for any given suitably regular initial data the problem posses a unique global-in-time classical solution which is uniformly bounded (see also Tao and Winkler [11]).
Xu, Zhang and Jin [13] studied the problem It is shown that under zero-flux boundary conditions, for any m > 0, the above problem admits a global bounded weak solution.They also discussed the large time behavior of solutions for the fast diffusion case, and showed that if 0 < m ≤ 1, for appropriately large µ, for any initial datum, the solution (u, v, w) goes to a steady (1, 1, 0) as t → ∞.
Zheng [15] investigated the Keller-Segel-Stokes system with nonlinear diffusion He proved that if m > 4/3, then for any sufficiently regular nonnegative initial data there exists at least one global bounded solution for system, which in view of the known results for the fluid-free system is an optimal restriction on m.Tao and Li [7] studied the chemotaxis-Navier-Stokes system They show that if p > 2, under appropriate assumptions on f and χ, for all sufficiently smooth initial data (n 0 , c 0 , u 0 ), the system has at least one global weak solution.The relevant equations have also been studied in [8,14].Liu and Li [6] studied the problem They proved that the problem admits a global bounded weak solution.
Li [5] considered an attraction-repulsion chemotaxis system with p-Laplacian diffusion Now we state our assumptions: In section 2, the boundary conditions become equivalent to This article is organized as follows: in Section 2, we prove some lemmas on the regularized problem of the system (1.1).In Section 3, the main result on the existence of a weak solution.

Regularized problem
We consider a regularized problem for solving system (1.1), where From ODE theory, (2.5) Tao and Winkler [10] stated that Thanks to ∂wε0 ∂ n ∂Ω = 0 and ∂vε ∂ n ∂Ω = 0 and using (2.4), we posses ∂wε ∂ n ∂Ω = 0. Considering the zero-flux boundary conditions of the system (2.1), the boundary conditions are equivalent to Based on a fixed point argument similar the one in [9] or [7], the local classical solution existence result of problem (2.1) can be proved.
Proof.According to (2.3), By the above inequality and the Young inequality, we have Multiplying the second equation in (2.1) by v ε , integrating over Ω, and using the Young inequality, we obtain 1 2 By (1.2) and (2.10), we obtain Using the Hölder inequality, we have . (2.16) From (1.2), the Sobolev imbedding theorem, and the Young inequality, we have Multiplying the second equation in (2.1) by v εt , integrating it over Ω, applying the Young inequality, and combining with (1.2), we obtain Similarly, multiplying the second equation in (2.1) by ∆v ε , integrating it over Ω, and using the Young inequality, we obtain The proof is complete.
2), we have where C is independent of ε.
Proof.we define where C 1 and C 2 are constants independent of k.Let

.45)
where p = p p−1 .It is obvious that Note that where and integrating over Ω, we obtain , and combining the Young inequality, we have The above inequality indicates that (2.49) The remaining part of the proof can be done in the same way as that in the proof of [6, Lemma 2.6], we omit the details.
, s for all T > 0, we have where C is independent of ε.