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{$\mathrm{◊}$} \begin{cases} u_{ t}=\Delta u -\chi \nabla \cdot (u\nabla v)+\xi \nabla \cdot (u\nabla w) & $\text{in }$\Omega\times (0,t^), \ 0=\Delta v+\alpha u-\beta v & $\text{in }$ \Omega\times (0,t^),\ 0=\Delta w+\gamma u-\delta w & $\text{in }$ \Omega\times (0,t^*),\ \end{cases} \end{equation} where $\mathrm{\Omega }$ is a smooth and bounded domain of ${\mathbb{R}}^2$, $t^{\ast }$ 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 $t\nearrow t^{\ast }$. The main result of this investigation is to provide an explicit lower bound for $t^{\ast }$ estimated in terms of ${\int }_{\mathrm{\Omega }}{u}_0^{2}d\mathbf{x}$ and attained by means of well–established techniques based on ordinary differential inequalities.