Global well-posedness to a multidimensional parabolic-elliptic-elliptic attraction-repulsion chemotaxis system

In this article we study the initial-boundary value problem for the attraction-repulsion chemotaxis system $$\displaylines{ u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w), \quad x\in\Omega,; t>0,\cr 0=\Delta v-\beta v+\alpha u, \quad x\in\Omega,\; t>0,\cr 0=\Delta w-\delta w+\gamma u, \quad x\in\Omega,\; t>0,\cr \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu} =\frac{\partial w}{\partial \nu}=0, \quad x\in\partial\Omega,\; t>0,\cr u(x,0)=u_0(x), \quad x\in\Omega, }$$ with homogenous Neumann boundary conditions in a multidimensional bounded domain \(\Omega\subset\mathbb{R}^N\) \( (1\leq N\leq4)\) with smooth boundary, where \(\chi\), \(\xi\), \(\alpha\), \(\beta\), \(\delta\) and \(\gamma\) are positive constants. We prove that under the assumption \(\chi\alpha=\xi\gamma\) the corresponding system possesses a unique global bounded classical solution in the cases \(N\leq 3\) or \(\lambda_0 \gamma\delta\xi \| u_0\|^{10/7}_{L^1(\Omega)} < \frac{1}{C_{GN}}\) and \(N=4\). Moreover, the large time behavior of solutions is also investigated. Specially, when \(\chi\alpha=\xi\gamma\), the solution of the system converges to \((\bar{u}_0,\frac\alpha\beta\bar{u}_0,\frac\gamma\delta\bar{u}_0)\) exponentially if \(\|u_0\|_{L^\infty(\Omega)}\) is small. For more information see https://ejde.math.txstate.edu/Volumes/2025/26/abstr.html