Global in-time behavior for the semilinear nonlocal heat exchanger system with different diffusion orders

We consider a semilinear nonlocal heat exchanger system whose two equations may have different fractional diffusion orders. Although these orders need not coincide, both components have the same leading diffusion for large time. Its order is $σ_*:=\min\{σ_1,σ_2\}$, while its coefficient depends on w