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

Abstract

In this paper we study classical solutions to the zero--flux attraction--repulsion chemotaxis--system \begin{equation}\label{ProblemAbstract} \tag{$◊$} \begin{cases} u_{ t}=Δu -χ\nabla \cdot (u\nabla v)+ξ\nabla \cdot (u\nabla w) & \textrm{in }Ω\times (0,t^*), \\ 0=Δv+αu-βv & \textrm{in } Ω\times (0,t^*),\\ 0=Δw+γu-δw & \textrm{in } Ω\times (0,t^*),\\ \end{cases} \end{equation} where $\Omega$ 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 $u(\mathbf{x},0)=u_0\in C^0(\bar{\Omega })$, 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 ${\int }_{\Omega }{u}_0^{2}d\mathbf{x}$ and attained by means of well--established techniques based on ordinary differential inequalities.