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.
@article{TretonBlowUp24,
title = {Blow-{{Up}} vs. {{Global Existence}} for a {{Fujita-Type Heat Exchanger System}}},
author = {Tr{\'e}ton, Samuel},
year = {2024},
journal = {SIAM Journal on Mathematical Analysis},
pages = {2191--2212},
publisher = {{Society for Industrial and Applied Mathematics}},
issn = {0036-1410},
doi = {10.1137/23M1587440},
abstract = {We analyze a reaction-diffusion system on \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}
}
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 recover the so-called field-road diffusion model as the hydrodynamic limit
of an interacting particle system. The former consists of two parabolic PDEs posed on two sets of different dimensions
(a "field" and a "road" in a population dynamics context), and coupled through exchange terms between the
field's boundary and the road.
The latter stands as a Symmetric Simple Exclusion Process (SSEP): particles evolve on two microscopic lattices following
a Markov jump process, with the constraint that each site cannot host more than one particle at the same time.
The system is in contact with reservoirs that allow to create or remove particles at the boundary sites. The dynamics of
these reservoirs are slowed down compared to the diffusive dynamics, to reach the reactions and the boundary conditions
awaited at the macroscopic scale.
This issue of bridging two spaces of different dimensions is, as far as we know, new in the hydrodynamic limit context,
and raises perspectives towards future related works.
@misc{AlfaroBridging24,
title = {Bridging Bulk and Surface: {{An}} Interacting Particle System towards the Field-Road Diffusion Model},
shorttitle = {Bridging Bulk and Surface},
author = {Alfaro, Matthieu and Mourragui, Mustapha and Tr{\'e}ton, Samuel},
year = {2024},
month = jun,
number = {arXiv:2406.14093},
eprint = {2406.14093},
primaryclass = {math},
publisher = {arXiv},
doi = {10.48550/arXiv.2406.14093},
abstract = {We recover the so-called field-road diffusion model as the hydrodynamic limit of an interacting particle
system. The former consists of two parabolic PDEs posed on two sets of different dimensions (a "field" and a "road" in a
population dynamics context), and coupled through exchange terms between the field's boundary and the road. The latter
stands as a Symmetric Simple Exclusion Process (SSEP): particles evolve on two microscopic lattices following a Markov
jump process, with the constraint that each site cannot host more than one particle at the same time. The system is in
contact with reservoirs that allow to create or remove particles at the boundary sites. The dynamics of these reservoirs
are slowed down compared to the diffusive dynamics, to reach the reactions and the boundary conditions awaited at the
macroscopic scale. This issue of bridging two spaces of different dimensions is, as far as we know, new in the
hydrodynamic limit context, and raises perspectives towards future related works.}
}
