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Positivity and long-term behaviour of a diffusion model with measure-valued nonlocal reaction term

Xiao Yang, Qiyao Peng, Sander C. Hille

TL;DR

The paper analyzes a 1D diffusion equation on $\mathbb{R}$ with a nonlocal, measure-valued reaction term localized at the origin, where the reaction intensity depends on solution values at symmetric boundary points. By leveraging Laplace transform techniques and a heat-kernel transfer-operator framework, it derives conditions under which the exterior solution remains nonnegative and establishes convergence to a constant steady state when a small feedback parameter is used. The key contributions are explicit positivity criteria for the uptake parameter $a$ (namely $0\le a\le e^{-1}$), a renewal-structure analysis, and a rigorous link between inverse Laplace transforms and pointwise positivity outside the observation region, all supported by numerical experiments. The results provide a rigorous foundation for positivity and long-time behavior in nonlocal diffusion-control models with Dirac-type reactions, with potential applications to sensor-driven interfacial systems.

Abstract

The behaviour is investigated of solutions to a diffusion equation on the real line with nonlocal and singular reaction term, i.e., given by a Dirac source or sink at the origin. It gives a simplified representation of for example a control system that senses concentration at a distance, but "intervenes" at the origin. Positivity of solutions (for positive initial conditions) cannot be guaranteed for all parameter settings in the model. We determine a parameter regime and conditions on the positive initial condition in terms of monotonicity and symmetry, that do allow us to conclude the positivity of the solution for all time. In addition, we provide conditions that ensure convergence of the system to a constant steady state (pointwise), outside the region of observation. Technically, we extensively use Laplace transform arguments to achieve these results.

Positivity and long-term behaviour of a diffusion model with measure-valued nonlocal reaction term

TL;DR

The paper analyzes a 1D diffusion equation on with a nonlocal, measure-valued reaction term localized at the origin, where the reaction intensity depends on solution values at symmetric boundary points. By leveraging Laplace transform techniques and a heat-kernel transfer-operator framework, it derives conditions under which the exterior solution remains nonnegative and establishes convergence to a constant steady state when a small feedback parameter is used. The key contributions are explicit positivity criteria for the uptake parameter (namely ), a renewal-structure analysis, and a rigorous link between inverse Laplace transforms and pointwise positivity outside the observation region, all supported by numerical experiments. The results provide a rigorous foundation for positivity and long-time behavior in nonlocal diffusion-control models with Dirac-type reactions, with potential applications to sensor-driven interfacial systems.

Abstract

The behaviour is investigated of solutions to a diffusion equation on the real line with nonlocal and singular reaction term, i.e., given by a Dirac source or sink at the origin. It gives a simplified representation of for example a control system that senses concentration at a distance, but "intervenes" at the origin. Positivity of solutions (for positive initial conditions) cannot be guaranteed for all parameter settings in the model. We determine a parameter regime and conditions on the positive initial condition in terms of monotonicity and symmetry, that do allow us to conclude the positivity of the solution for all time. In addition, we provide conditions that ensure convergence of the system to a constant steady state (pointwise), outside the region of observation. Technically, we extensively use Laplace transform arguments to achieve these results.

Paper Structure

This paper contains 25 sections, 49 theorems, 176 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Mathematical assumptions
  3. Question of positivity of solution in relation to a control problem
  4. Non-dimensionalization of the model
  5. Structure of the manuscript
  6. Overview of main results
  7. Employing Laplace transform for proving positivity
  8. Positivity of solutions on part of the domain
  9. Pointwise positivity on the boundary suffices
  10. Positivity of the sum: Lg
  11. A Renewal Equation for $u_+$
  12. An expression for $u_+$ in the $s$-domain
  13. Outline of the approach to proving positivity of $u_+$
  14. Analysis of $\tilde{p}_a(s,\beta)$ by related delay equation
  15. Further analysis of $P_a(s,\beta)$ and $\tilde{P}_a(s,\beta$) and their inverse Laplace transforms
  16. ...and 10 more sections

Key Result

Theorem 2.1

Let $a\in\mathbb{R}$. System $($eq: system considered u$)$ -- $($def:Psi new$)$ has a unique mild solution $u\in C^0(\mathbb{R}_0^+,H^1(\mathbb{R}))$, provided $u_{0}\in H^1(\mathbb{R})$ and $\Phi\in L^\infty_{\mathrm{loc}}(\mathbb{R}_0^+)$. Moreover, $u$ depends continuously on the initial conditio

Figures (14)

  • Figure 1.1: Schematic of immune cells migrating to wound region and cancer cell metastasis via the bloodstream. The cells enter the blood vessels from one part of the body and exit from the other part. Projecting onto x-space, the blood vessels and the trajectory of cells entering and exiting the blood vessels can be simplified to a one-dimensional model, as in System $($\ref{['eq: system considered u']}$)$--$($\ref{['def:Psi new']}$)$, where $u$ then stands for the cell density in these two biological examples.
  • Figure 1.2: Feedback block diagram that represents the dynamics of $u_+$ in s-domain in Section \ref{['sec:expression of u+ s-domain']}. For the purpose of expressing the control system in a general setting, we use the symbols that are not relevant to our study.
  • Figure 3.1: Flowchart of applying Laplace transform, where $\mathcal{L}$ represents Laplace transform and $\mathcal{L}^{-1}$ is inverse Laplace transform.
  • Figure 4.1: Regions of $a$ and $\beta$ regarding the positivity of $p_a(t,\beta)$ determined by the existence of the positive real root in Equation $($\ref{['Eq_P_a_derivative']}$)$ is shown. Thus, the positivity of $u_+$ is rejected or possible accordingly. The regions are separated by the critical curve $a\beta - a + 1 =0$.
  • Figure 4.2: Change of shape of $\tilde{p}_a(t,0)$ with varying $a$. The function has been plotted using the exact expression $($\ref{['eq:lapinvpsub']}$)$. Left: when $a\leqslant0$, particularly, $a\in\{0,0.5, -1\}$. Right: when $a>0$, in particular $a \in \{0.25, 1, 2\}$. Note that for $a\in\{1,2\}$, $\tilde{p}_a$ is oscillatory and can become negative.
  • ...and 9 more figures

Theorems & Definitions (95)

  • Theorem 2.1: Well-posedness
  • proof
  • Theorem 2.2: Positivity under monotonicty condition
  • Theorem 2.3
  • Proposition 2.4: Positivity under symmetry condition
  • Remark 2.5
  • Theorem 3.1: Uniqueness Theorem
  • Corollary 3.2
  • Theorem 3.3: Paley-Wiener
  • Proposition 3.4
  • ...and 85 more