Non-simultaneous blow-up for a system with local and non-local diffusion

We study the possibility of non-simultaneous blow-up for positive solutions of a coupled system of two semilinear equations, $u_t = J*u-u+ u^\alpha v^p$, $v_t =\Delta v^+u^qv^\beta$, $p, q, \alpha, \beta>0$ with homogeneous Dirichlet boundary conditions and positive initial data. We also give the blow-up rates for non-simultaneous blow-up.