Fujita-type Phenomenon of the Nonlocal Diffusion Equations with Localized Source

Abstract:

      In this paper, we investigate the Cauchy problem for the nonlocal diffusion system with localized source $u_t=J*u-u+a(x)v^{p}$, $v_t=J*v-v+a(x)u^q$. We first prove that the Fujita curve is $(pq)_c=1+\max \{p+1,q+1\}$ based on whether there exist global solutions, that is, if $1(pq)_c$, there exist both global and non-global solutions to the problem. Furthermore, we establish the secondary critical curve on the space-decay of initial value at infinity.