Non-extinction of solutions to a fast diffusion system with nonlocal sources

In this short article, we give a positive answer to the problem proposed by Zheng et al [5], and show that the fast diffusion system $$\displaylines{ u_t=\hbox{div}(|\nabla u|^{p-2}\nabla u) +\int_\Omega v^\alpha\,dx, \cr v_t =\hbox{div}(|\nabla v|^{q-2}\nabla v) +\int_\Omega u^\beta\,dx }$$ under homogeneous Dirichlet boundary condition admits at least one non-extinction solution when $\alpha\beta<(p-1)(q-1)$ and the initial data are strictly positive