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Lecture notes: Biological propagation via reaction-diffusion equations with nonlocal diffusion and free boundary

Yihong Du

TL;DR

This work analyzes propagation in a one-dimensional nonlocal reaction-diffusion system with free boundaries and a KPP-type nonlinearity. It develops a comprehensive framework combining a maximum principle, comparison methods, and an auxiliary fixed-domain problem to establish global existence and uniqueness of solutions, and to characterize the spreading-vanishing dichotomy via spectral properties on finite intervals. By introducing and analyzing semi-wave solutions, it connects front speeds to traveling-wave theory, yielding precise upper and lower bounds and convergence results, and it finally derives sharp acceleration rates for front propagation under various tail behaviors of the nonlocal kernel $J$. The results extend to weakly non-symmetric kernels, offering a robust toolkit for predicting propagation dynamics in nonlocal free-boundary models with broad applicability in ecology and epidemiology.

Abstract

These notes are based on the lectures given in a mini-course at VIASM (Vietnam Institute for Advanced Study in Mathematics) 2025 Summer School. They give a brief account of the theory (with detailed proofs) for propagation governed by a nonlocal reaction-diffusion model with free boundaries in one space dimension. The main part is concerned with a KPP reaction term, though the basic results on the existence and uniqueness of solutions as well as on the comparison principles are for more general situations. The contents are mostly taken from published recent works of the author with several collaborators, where the kernel function was assumed to be symmetric: J(x)=J(-x). When J(x) is not symmetric, significant differences may arise in the dynamics of the model, as shown in several preprints quoted in the references at the end of these notes, but many of the existing techniques can be easily extended to cover the "weakly non-symmetric case", and this is done here with all the necessary details.

Lecture notes: Biological propagation via reaction-diffusion equations with nonlocal diffusion and free boundary

TL;DR

This work analyzes propagation in a one-dimensional nonlocal reaction-diffusion system with free boundaries and a KPP-type nonlinearity. It develops a comprehensive framework combining a maximum principle, comparison methods, and an auxiliary fixed-domain problem to establish global existence and uniqueness of solutions, and to characterize the spreading-vanishing dichotomy via spectral properties on finite intervals. By introducing and analyzing semi-wave solutions, it connects front speeds to traveling-wave theory, yielding precise upper and lower bounds and convergence results, and it finally derives sharp acceleration rates for front propagation under various tail behaviors of the nonlocal kernel . The results extend to weakly non-symmetric kernels, offering a robust toolkit for predicting propagation dynamics in nonlocal free-boundary models with broad applicability in ecology and epidemiology.

Abstract

These notes are based on the lectures given in a mini-course at VIASM (Vietnam Institute for Advanced Study in Mathematics) 2025 Summer School. They give a brief account of the theory (with detailed proofs) for propagation governed by a nonlocal reaction-diffusion model with free boundaries in one space dimension. The main part is concerned with a KPP reaction term, though the basic results on the existence and uniqueness of solutions as well as on the comparison principles are for more general situations. The contents are mostly taken from published recent works of the author with several collaborators, where the kernel function was assumed to be symmetric: J(x)=J(-x). When J(x) is not symmetric, significant differences may arise in the dynamics of the model, as shown in several preprints quoted in the references at the end of these notes, but many of the existing techniques can be easily extended to cover the "weakly non-symmetric case", and this is done here with all the necessary details.

Paper Structure

This paper contains 29 sections, 39 theorems, 484 equations.

Table of Contents

  1. Maximum principle and comparison results
  2. A maximum principle
  3. An example
  4. A comparison result for a scalar nonlocal free boundary problem
  5. Existence and uniqueness
  6. An auxiliary initial boundary value problem
  7. Proof of Theorem \ref{['Thm22']}
  8. Spreading-vanishing dichotomy and criteria
  9. The associated problem over a fixed spatial interval
  10. Proof of Theorem \ref{['thm2']}
  11. Proof of Theorem \ref{['thm3']}
  12. Semi-wave solutions
  13. A maximum principle and its first application
  14. A perturbed semi-wave problem
  15. A dichotomy between semi-waves and traveling waves
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $T, h_0>0$, $g, h\in C([0,T])$ satisfy $g(t)<h(t)$ and $-g (0)=h(0)=h_0$. Denote $D_T:=\{(t,x): t\in (0, T],\; g(t)<x<h(t)\}$ and suppose that for $i,j\in \{1,2, ..., n\}$, $\phi_i$, $\partial_t\phi_i\in C(\overline D_T)$, $d_i$, $c_{ij}\in L^\infty(D_T)$, ${\color{red}d_i\geq 0}$, and where $\mathcal{L}_i$ is given by Li with every $J_i$$(i=1,..., n)$ satisfying (J). Then the following con

Theorems & Definitions (65)

  • Theorem 1.1: Maximum Principle
  • proof
  • Corollary 1.2
  • proof
  • Theorem 1.3
  • proof
  • Theorem 2.1
  • Lemma 2.2
  • Theorem 3.1: Spreading-vanishing dichotomy
  • Theorem 3.2: Spreading-vanishing criteria
  • ...and 55 more