Explicit lower bound of blow--up time for an attraction--repulsion chemotaxis system

In this paper we study classical solutions to the zero--flux attraction--repulsion chemotaxis--system \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} \begin{cases} u_{ t}=\Delta u -\chi \nabla \cdot (u\nabla v)+\xi \nabla \cdot (u\nabla w) & \textrm{in }\Omega\times (0,t^*), \\ 0=\Delta v+\alpha u-\beta v & \textrm{in } \Omega\times (0,t^*),\\ 0=\Delta w+\gamma u-\delta w & \textrm{in } \Omega\times (0,t^*),\\ \end{cases} \end{equation} where $\Omega$ is a smooth and bounded domain of $\mathbb{R}^2$, $t^*$ is the blow--up time and $\alpha,\beta,\gamma,\delta,\chi,\xi$ are positive real numbers. From the literature it is known that under a proper interplay between the above parameters and suitable smallness assumptions on the initial data $u({\bf x},0)=u_0\in C^0(\bar{\Omega})$, system \eqref{ProblemAbstract} has a unique classical solution which becomes unbounded as $t\nearrow t^*$. The main result of this investigation is to provide an explicit lower bound for $t^*$ estimated in terms of $\int_\Omega u_0^2 d{\bf x}$ and attained by means of well--established techniques based on ordinary differential inequalities.