A refined criterion and lower bounds for the blow--up time in a parabolic--elliptic chemotaxis system with nonlinear diffusion

This paper deals with unbounded solutions to the following zero–flux chemotaxis system \begin{equation}\label{ProblemAbstract} \tag{$\mathrm{◊}$} \begin{cases} % about u u_t=\nabla \cdot [(u+\alpha)^{m_1-1} \nabla u-\chi u(u+\alpha)^{m_2-2} \nabla v] & (x,t) \in \Omega \times (0,T_{max}), \[1mm] % about v 0=\Delta v-M+u & (x,t) \in \Omega \times (0,T_{max}), \end{cases} \end{equation} where $\alpha >0$, $\mathrm{\Omega }$ is a smooth and bounded domain of ${\mathbb{R}}^n$, with $n\ge 1$, $t\in (0,T_{max})$, where $T_{max}$ the blow-up time, and $m_1,m_2$ real numbers. Given a sufficiently smooth initial data $u_0:=u(x,0)\ge 0$ and set $M:=\frac{1}{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}{\int }_{\mathrm{\Omega }}{u}_0(x)dx$, from the literature it is known that under a proper interplay between the above parameters $m_1,m_2$ and the extra condition ${\int }_{\mathrm{\Omega }}v(x,t)dx=0$, system \eqref{ProblemAbstract} possesses for any $\chi >0$ a unique classical solution which becomes unbounded at $t\nearrow T_{max}$. In this investigation we first show that for $p_0>\frac{n}{2}(m_2-m_1)$ any blowing up classical solution in $L^{\mathrm{\infty }}(\mathrm{\Omega })$–norm blows up also in $L^{p_0}(\mathrm{\Omega })$–norm. Then we estimate the blow–up time $T_{max}$ providing a lower bound $T$.