pper and Lower Bounds for Blow-up Times in Coupled Reaction--Diffusion Systems with Nonlocal Sources

In this paper, we consider the following reaction-diffusion system \begin{equation*} \begin{cases} u_t=Δu+F\bigl(x,t,u,v,V(t)\bigr), & (x,t)\in\Om\times(0,T), v_t=Δv+G\bigl(x,t,u,v,U(t)\bigr), & (x,t)\in\Om\times(0,T), \mathcal B u=\mathcal B v=0, & (x,t)\in\partial\Om\times(0,T), u(x,0)=u_0(x),\quad v(x,0)=v_0(x), & x\in\Om, \end{cases} \end{equation*} where \begin{equation*} U(t)=\int_{\Om}u(x,t)\dd x,\qquad V(t)=\int_{\Om}v(x,t)\dd x, \end{equation*} and the boundary operator $\mathcal{B}$ is either $${\mathcal{B}}_{D}w=w\text{or}{\mathcal{B}}_{N}w={\mathrm{\partial }}_{\nu }w,$$ $\text{Om}\subset {\mathbb{R}}^N$ be a bounded domain with sufficiently smooth boundary, $N\ge 1$, and $\nu$ is the outward unit normal on $\mathrm{\partial }\text{Om}$. Under certain conditions, we obtain the upper and lower bounds for the blowup time of the solution to the system.