A note on global in-time behavior for the semilinear nonlocal heat exchanger system

We mainly study global in-time asymptotic behavior for the nonlocal reaction-diffusion system with fractional Laplacians which models dispersal of individuals between two exchanging environments for its diffusive components and incorporates the Fujita-type power nonlinearities for its reactive components. We derive a global in-time existence result in the super-critical case, and large time asymptotic profiles of global in-time solutions in the general $L^m$ framework. As a byproduct, the sharp lower bound estimates of lifespan for local in-time solutions in the sub-critical and critical cases are determined. These results extend the existence part of [S. Tr\'eton, SIAM J. Math. Anal. (2024)].