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

TL;DR

This paper investigates the long-term behavior of a nonlocal reaction-diffusion system with fractional Laplacians, establishing global existence, asymptotic profiles, and lifespan estimates across different nonlinear regimes.

Contribution

It extends previous results by providing comprehensive global existence and asymptotic analysis for a fractional nonlocal heat exchanger system with Fujita-type nonlinearities.

Findings

Global in-time solutions exist in the super-critical case.

Asymptotic profiles of solutions are characterized for large times.

Lifespan estimates are obtained for sub-critical and critical cases.

Abstract

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)].