Global existence and blow-up for non-Newton polytropic filtration system with nonlocal source

Abstract

This paper deals the global existence and blow-up properties of the following non-Newton polytropic filtration system with nonlocal source, $$u_t-\triangle_{m,p}u=a\int_{\Omega}v^\alpha (x,t)dx, v_t-\triangle_{n,q}v=b\int_{\Omega}u^\beta (x,t)dx.$$ Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time depends on the initial data and the relations between $\alpha\beta$ and $mn(p-1)(q-1)$. In the special case, $\alpha=n(q-1), \beta=m(p-1)$, we also give a criteria for the solution exists globally or blows up in finite time, which depends on $ab$ and $\zeta(x), \vartheta(x)$ defined in our main results. doi:10.1017/S1446181108000242