A new criterion to a two-chemical substances chemotaxis system with critical dimension

We mainly investigate the global boundedness of the solution to the following system, \begin{document}$\begin{align*}\begin{cases}u_t = Δ u-χ\nabla·(u\nabla v) &\text{ in }Ω×\mathbb R^+,\\v_t = Δ v-v+w &\text{ in }Ω×\mathbb R^+,\\w_t = Δ w-w+u &\text{ in }Ω×\mathbb R^+,\end{cases}\end{align*}$ \end{document} under homogeneous Neumann boundary conditions with nonnegative smooth initial data in a smooth bounded domain $Ω\subset \mathbb{R}^n$ with critical space dimension $n = 4$. This problem has been considered by K. Fujie and T. Senba in [ 5 ]. They proved that for the symmetric case the condition $\int_\Omega {u_0 3 ], we give a new criterion for global boundedness of the solution. As a byproduct, we obtain a simplified proof for one of the main results in [ 5 ].

under homogeneous Neumann boundary conditions with nonnegative smooth initial data in a smooth bounded domain Ω ⊂ R n with critical space dimension n = 4. This problem has been considered by K. Fujie and T. Senba in [5].
They proved that for the symmetric case the condition Ω u 0 < (8π) 2 χ yields global boundedness, where u 0 is the instal data for u. In this paper, inspired by some new techniques established in [3], we give a new criterion for global boundedness of the solution. As a byproduct, we obtain a simplified proof for one of the main results in [5].
A detailed introduction of this model could be found in K. Fujie and T. Senba in [5] and for more discussion related to chemotaxis models, we refer the reader to survey papers [2,7,8]. In this paper, we intend to give an extension as well as a simplified proof for one of the main results in [5] in the critical case n = 4. In particular, we establish the following result.
If in addition we assume that either

Remark 1.
If Ω = B R (0), u 0 , v 0 , w 0 are radially symmetric and Ω u 0 < (8π) 2 χ , K. Fujie and T. Senba have shown that (1) admits a unique solution satisfying (see [5,Lemma 7 which implies that u is uniformly integrable. Therefore, our result can be considered as a new and shorter proof of [5,Theorem 1.3]. Without the radially symmetric assumption, we show that if Ω u 0 dx < ε 0 with ε 0 small enough, the solution is globally bounded. Here ε may be less than (8π) 2 2χ . It is still open that whether the condition Ω u 0 dx < (8π) 2 2χ can ensure the global boundedness for solution of (1) when Ω is general domain(see [5,Remark 1.5]).
2. preliminaries. In this part, we give several lemmas which are fatal to our proof, we will use the symbol · p to denote the L p (Ω) norm for simplicity. The first one is the optimal sobolev regularity lemma. Instead of using [6, Theorem 3.1] directly, we adopt [4, Lemma 2.5] (with appropriate modifications) here which is more effective for our purpose.
Proof of Theorem 1.1. Since the local existence and global boundedness under uniform L p -prior estimate of u has been established in [5] (Proposition 4.1 and Lemma 5.4), we only need to show that there exists p > 1 such that sup 0<t<Tmax u(·, t) p is bounded. It is routine to check that u L 1 (Ω) , v L 1 (Ω) and w L 1 (Ω) are uniformly bounded in t. Hence, throughout the rest of this paper, we always assume that C is a positive constant which may depend on sup 0<t<Tmax ( u L 1 (Ω) + v L 1 (Ω) + w L 1 (Ω) ) and change from place to place. Multiplying the first equation by 1 p u p−1 , we obtain that ∂ ∂t Using Gagliardo-Nirenberg interpolation inequality and Young's inequality, we can estimate Ω u p |∇v| 2 dx as follows: