Blow-up vs. global existence for a Fujita-type Heat exchanger system

TL;DR

This paper studies a coupled reaction-diffusion system modeling dispersal between environments with Fujita-type growth, identifying conditions for blow-up or global existence and analyzing the system's diffusive behavior.

Contribution

It introduces a novel coupled model with diffusion and Fujita-type reactions, analyzing its blow-up and global existence thresholds, which has not been previously studied.

Findings

Solutions to the diffusive problem converge exponentially to uncoupled equations.

Identified the critical exponent separating blow-up from global existence.

Analyzed the impact of coupling on the system's long-term behavior.

Abstract

We analyze a reaction-diffusion system on $\mathbb{R}^{N}$ which models the dispersal of individuals between two exchanging environments for its diffusive component and incorporates a Fujita-type growth for its reactive component. The originality of this model lies in the coupling of the equations through diffusion, which, to the best of our knowledge, has not been studied in Fujita-type problems. We first consider the underlying diffusive problem, demonstrating that the solutions converge exponentially fast towards those of two uncoupled equations, featuring a dispersive operator that is somehow a combination of Laplacians. By subsequently adding Fujita-type reaction terms to recover the entire system, we identify the critical exponent that separates systematic blow-up from possible global existence.