I am currently working on the diffusive aspect of the reaction-diffusion equations on the field-road space ℝ^{N-1}×ℝ_{+} with N≥2 (introduced by H.Berestycki, J.-M.Roquejoffre and L.Rossi in 2012). More precisely, I am interesting in the solution (u,v) = ( u(t,x) , v(t,x,y) ) of the following problem:
starting from an initial datum (u_{0},v_{0}) = ( u_{0}(x) , v_{0}(x,y) ) bounded and integrable (a step for example).
One calls "field the domain ℝ^{N-1}×ℝ_{+} and "road" the field's boundary, ie. ℝ^{N-1}×{0}.
The function v stands for the individuals density on the field while u represents the individuals density on the road.
Individuals diffuse with a coefficient d on the field and with a coefficient D on the road with, typically, D>d.
Second line of the previous system is called exchange condition. This boundary condition is of sourced-Robin type and is in the core of the model since it allows to manage the behavior of individuals which arrive at the boundary of the field (bouncing or exiting towards the road) and immigration from the road.
Taking the associated linear system, ie. f≡0, one may show that the mass of the population is preserved over time; we thus get a purely diffusive system.
Knowing that the action of the diffusion in the whole space ℝ^{N} on a reasonable datum (bounded and integrable) provokes the extinction of the population by spreading individuals (uniform convergence of the solution toward 0 as t→+∞), one wonders to what extend the inclusion of the road (ie. a fast diffusion channel) may impact the rate of population extinction.
The simulations here based on microscopic indivual motions are a good introduction to well understand the dynamic of the field-road model.
In this presentation, we'll discuss finite-time explosion phenomena arising for certain superlinear reaction-diffusion problems.
We'll start with an introduction to the seminal results of Japanese mathematician H.Fujita concerning the altered Heat equation by adding the unknown raised to the power 1+p (p>0) in the second member. In his 1966 work, Fujita highlighted a critical exponent for p, marking the threshold between systematic explosion and possible global existence of solutions. This distinction is based on an equilibrium ratio between two remarkable algebraic quantities associated with the reactive (growth) and diffusive (mass scattering) parts of the equation.
We will then broaden our perspective by introducing a "heat exchanger" system where the unknowns are coupled by a diffusion mechanism, while integrating over-linear and non-coupling reactions as previously stated. A frequency analysis of the purely diffusive heat exchanger will enable us to estimate its "scattering intensity", leading to the main results of the talk concerning the systematic explosion and possible global existence of solutions of such a semilinear system.
This work is a first step towards extending Fujita-type problems to diffusion-coupled systems, and raises several open questions, including the exploration of more complex diffusive mechanisms...
In this talk, we examine mathematical models that describe the diffusion and exchange of individuals across spatial domains.
We begin with the field-road model, emphasizing its biological foundations and its importance in understanding fast diffusion channels in population dynamics and ecology. Following this, we explain how to derive the explicit solutions for the field-road model and provide estimates on the asymptotic decay rate of these solutions.
This analytical framework paves the way for exploring non-linear issues, including Fujita-type blow-up phenomena, which we explain by outlining the key concepts involved.
We then turn to the Heat-exchanger model, which serves as a tractable first approach for deriving some coupled-by-diffusion Fujita-type systems. We proceed to characterize the purely diffusive version of this model. With this foundation, we tackle the question of blow-up vs. global existence that arises when incorporating super-linear Fujita-type reaction terms.
The presentation concludes with insights into a stochastic simple exclusion process for the field-road diffusion model. We provide a brief overview of the mechanics of this particle system, drawing parallels with the canonical example of the Heat equation.
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.