We consider the linear field-road system, a model for fast diffusion channels in population dynamics and ecology. This system takes the form of a system of PDEs set on domains of different dimensions, with exchange boundary conditions. Despite the intricate geometry of the problem, we provide an explicit expression for its fundamental solution and for the solution to the associated Cauchy problem. The main tool is a Fourier (on the road variable)/Laplace (on time) transform. In addition, we derive estimates for the decay rate of the L∞ norm of these solutions.
@article{AlfaroFieldRoad23,
title = {The Field-Road Diffusion Model: {{Fundamental}} Solution and Asymptotic Behavior},
shorttitle = {The Field-Road Diffusion Model},
author = {Alfaro, Matthieu and Ducasse, Romain and Tr{\'e}ton, Samuel},
year = {2023},
journal = {Journal of Differential Equations},
volume = {367},
pages = {332--365},
issn = {0022-0396},
doi = {10.1016/j.jde.2023.05.002},
abstract = {We consider the linear field-road system, a model for fast diffusion channels in population dynamics and ecology. This system takes the form of a system of PDEs set on domains of different dimensions, with exchange boundary conditions. Despite the intricate geometry of the problem, we provide an explicit expression for its fundamental solution and for the solution to the associated Cauchy problem. The main tool is a Fourier (on the road variable)/Laplace (on time) transform. In addition, we derive estimates for the decay rate of the $L^{\infty}$ norm of these solutions.},
keywords = {Decay rate, Diffusion, Exchange boundary conditions, Field-road model, Fundamental solution}
}
We analyze a reaction-diffusion system on ℝN 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.
@misc{TretonBlowup23,
title = {Blow-up vs. Global Existence for a {{Fujita-type Heat}} Exchanger System},
author = {Tr{\'e}ton, Samuel},
year = {2023},
month = jul,
number = {arXiv:2307.00843},
eprint = {2307.00843},
primaryclass = {math},
publisher = {{arXiv}},
doi = {10.48550/arXiv.2307.00843},
urldate = {2023-07-04},
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.},
keywords = {Reaction-diffusion system, Fujita blow-up phenomena, Critical exponent, Global solutions, Heat exchanger system.}
}
