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,

$\begin{align*}\begin{cases}u_t = Δ u-χ\nabla·(u\nabla v) &\text{ in }Ω×\mathbb R^+,\\v_t = Δ v-v+w &\text{ in }Ω×\mathbb R^+,\\w_t = Δ w-w+u &\text{ in }Ω×\mathbb R^+,\end{cases}\end{align*}$

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 < \frac{(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].