Global existence and blow-up of solutions to a nonlocal reaction-diffusion system (Q1431416)

The authors of this interesting paper study a reaction-diffusion system with nonlocal sources \[ \begin{aligned} u_t-\triangle u=\int_{\Omega }u^{\alpha }(x,t)v^p(x,t)\,dx, &\quad x\in \Omega , \;t>0,\\ v_t-\triangle v=\int_{\Omega }u^{q }(x,t)v^{\beta }(x,t)\,dx, &\quad x\in \Omega , \;t>0,\\ u(x,t)=v(x,t)=0, &\quad x\in\partial\Omega , \;t>0,\\ u(x,0)=u_0(x), \quad v(x,0)=v_0(x) , &\quad x\in\overline\Omega, \end{aligned} \] where \(\Omega \) is a bounded domain in \(\mathbb R^N\) (\(N\geq 1\)) with sufficiently smooth boundary \(\partial\Omega\), \(\alpha ,\beta , p, q\) are positive numbers which ensure that the equations in this system are completely coupled with the nonlinear reaction terms. Under appropriate hypotheses the authors obtain that the solution either exists globally or blows up in finite time by making use of super and sub solution techniques. In the case when the solution blows up in finite time, it is shown that the blow-up set covers the whole domain, which is quite different from the results with local sources. The blow-up rate of the solutions is discussed as well.