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, $u_t=J\ast u-u+u^{\alpha }{v}^p$, $v_t=\mathrm{\Delta }{v}^{+}{u}^q{v}^{\beta }$, $p,q,\alpha ,\beta >0$ with homogeneous Dirichlet boundary conditions and positive initial data. We also give the blow-up rates for non-simultaneous blow-up.