Global boundedness in a quasilinear two-species attraction-repulsion chemotaxis system with two chemicals (Q2083289)

The authors study the quasilinear attraction-repulsion chemotaxis system with two species and two chemicals \[ \begin{cases} u_t = \nabla \cdot (D_1(u) \nabla u) - \nabla \cdot(\Phi_1(u) \nabla v), & \quad (x,t) \in \Omega \times (0,\infty),\\ 0 = \Delta v - v + w^{\gamma_1}, & \quad (x,t) \in \Omega \times (0,\infty), \\ w_t = \nabla \cdot (D_2(w) \nabla w) + \nabla \cdot(\Phi_2(w) \nabla z), & \quad (x,t) \in \Omega \times (0,\infty),\\ 0 = \Delta z - z + u^{\gamma_2}, & \quad (x,t) \in \Omega \times (0,\infty), \end{cases} \] endowed with homogeneous Neumann boundary conditions and nonnegative initial data \(u_0, w_0 \in C(\overline{\Omega})\). Moreover, it is assumed that \(\Omega \subset \mathbb{R}^N\) is a bounded domain with smooth boundary with \(N \ge 2\) and that \(D_i, \Phi_i \in C^2([0,\infty))\) for \(i=1,2\) satisfy \(D_i(s) \ge (s+1)^{p_i}\) and \(\Phi_i(s) \ge 0\) for all \(s \ge 0\) as well as \(\Phi_i (s) \le \chi_i s^{q_i}\) for all \(s \ge s_0\) with some \(\gamma_i, \chi_i >0\), \(p_i, q_i \in \mathbb{R}\), and \(s_0 >1\). Here \(u\) and \(w\) denote the densities of two populations and \(v\) and \(z\) the concentration of two chemicals. While \(u\) is attracted by \(v\), which is produced by \(w\), \(w\) is repelled by \(z\), which is produced by \(u\). The authors prove the existence of a unique global classical solution to the above system provided that one of the following assumptions is fulfilled: (\(\gamma_1 < \frac{2}{N}\) and \(q_1 + \gamma_1 - p_1 < 1 + \frac{2}{N}\)) or (\(\gamma_2 < \frac{4}{N}\), \(\gamma_2 \le 1\), and \(q_2 + \gamma_2 - p_2 < 1 + \frac{4}{N}\)) or (\(q_i + \gamma_i - p_j \le 1+ \frac{2}{N}\) for \(i,j=1,2\)). The proof mainly relies on appropriate a priori estimates for \(u\) and \(w\).