Accelerating Propagation Induced by Slowly Decaying Initial Data for Nonlocal Reaction-Diffusion Equations in Cylinder Domains

Abstract

This paper investigates the phenomenon of accelerating propagation for nonlocal reaction-diffusion models with spatial and trait structure in a cylinder domain $\mathbb{R}\times \Omega$. Unlike previous studies focusing on exponentially decaying or compactly supported initial data, we consider initial functions that decay more slowly than any exponential function—such as algebraic or sub-exponential decay. By constructing a pair of super- and sub-solutions via the principal eigenfunction ${\psi }_0$ of the trait operator, we prove that the solution propagates with infinitely increasing speed in the spatial direction. Explicit upper and lower bounds for the locations of level sets are derived, illustrating how the decay rate of the initial data determines the acceleration profile. The results are extended to a more general model with space- and trait-dependent competition kernels under a boundedness assumption (H3). This work highlights the crucial role of slowly decaying tails in the initial distribution in driving accelerated invasion fronts, providing a theoretical foundation for assessing propagation risks in ecology and population dynamics.

1. Introduction

1.1. Model and Research Background

This paper is concerned with the spatial propagation of the nonlocal reaction-diffusion model in the cylinder domain $\mathbb{R}\times \Omega$ as follows

$$\left\{\begin{array}{c}{\partial }_{t}u(t,x,y)-{\Delta }_x{,}_{y}u(t,x,y)\hfill \\ =\left[1-g\left(y\right)-{\int }_{\Omega }K\left(y\right)u(t,x,y)dy\right]u(t,x,y),\hfill & \hfill & t>0,x\in \mathbb{R},y\in \Omega ,\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}=0,\hfill & \hfill & t>0,x\in \mathbb{R},y\in \partial \Omega ,\hfill \\ u(0,x,y)=u_0(x,y),\hfill & \hfill & x\in \mathbb{R},y\in \Omega ,\hfill \end{array}\right.$$

(1)

where $\Omega$ is a bounded domain in ${\mathbb{R}}^n$ with a smooth boundary $\partial \Omega$ and $\nu$ is the outward unit normal vector to $\mathbb{R}\times \partial \Omega$. This model can be used to study the internal competition among various phenotypes within a single population. Here $u(t,x,y)$ denotes the density of a population living in a continuous spatial variable $x\in \mathbb{R}$ and displaying a continuous bounded phenotypical trait $y\in \Omega$, where $\Omega$ is the set of all possible traits. The traits represent variations within species, such as rate of food intake, average litter size, flowering time, or age at maturity. The birth rate is given by $1-g\left(y\right)$, which takes into account the influence of natural selection on population survival. The nonlocal term ${\int }_{\Omega }K\left(y\right)u(t,x,y)dy$ indicates that the intra-specific competition occurs among individuals with any trait y at the location x, and $K\left(y\right)$ is the competitive ability of the individual with trait y. For more detailed biological background of (1), we refer to [1,2]. Moreover, the results in ([3] Theorem 2) showed that there is a unique solution of (1) for any nonnegative bounded initial data $u_0(x,y)\in L_{\infty }(\mathbb{R}\times \Omega )$, which is uniformly bounded in time.

We assume that $g\left(y\right)$ is a bounded and measurable function. Let ${\left\{{\lambda }_j\right\}}_{j=0}^{+\infty }$ be the eigenvalues of the operator $-{\Delta }_y+g\left(y\right)$ under homogeneous Neumann boundary condition on $\partial \Omega$, with ${\lambda }_0<{\lambda }_1⩽{\lambda }_2⩽\cdots ⩽{\lambda }_n⩽\cdots$ for $n\in \mathbb{N}$. Then first eigenvalue ${\lambda }_0$ is simple and real and corresponds to a positive eigenfunction ${\psi }_0\left(y\right)$, which implies that

$$\left\{\begin{array}{cc}[-{\Delta }_y+g\left(y\right)]{\psi }_0\left(y\right)={\lambda }_0{\psi }_0\left(y\right),\hfill & y\in \Omega ,\hfill \\ {\displaystyle \frac{\partial {\psi }_0\left(y\right)}{\partial \nu }}=0,\hfill & y\in \partial \Omega .\hfill \end{array}\right.$$

In this paper, we assume that

(H1) 

$g\in L_{\infty }(\Omega ),{\lambda }_0<1$, $K\in L_2(\Omega ),K\left(y\right)⩾0,K\left(y\right)\not\equiv 0,y\in \Omega$.

Under the assumption (H1), Berestycki et al. [4] obtained the existence and uniqueness of the cylinder front solution ${\varphi }_c(x-ct,y)$ of (1) for $c⩾c^{*}$ by applying spectral expansion and separation of variables, and showed that ${\varphi }_c(x-ct,y)$ must be in the form of $V_c(x-ct){\psi }_0\left(y\right)$, where $V_c$ satisfies

$$\left\{\begin{array}{c}{V}_c^{″}+c{V}_c^{\prime }+\left(1-{\lambda }_0-V_c{\int }_{\Omega }{\psi }_0\left(y\right)K\left(y\right)dy\right)V_c=0,\xi \in \mathbb{R},\hfill \\ V_c(+\infty )=0,V_c(-\infty )=(1-{\lambda }_0){\left[{\int }_{\Omega }{\psi }_0\left(y\right)K\left(y\right)dy\right]}^{-1}>0.\hfill \end{array}\right.$$

Under some additional assumptions on $K\left(y\right)$, it was also proven in [4] that the minimal speed $c^{*}$ is the spreading speed of the solution with a compactly supported initial data. Recently, Li et al. [3] obtained the locally exponential stability of the cylinder front solution in an exponentially weighted space. For the Cauchy problem (1), it was shown in [3] that the boundedness and the long time behavior of the solution are determined by that of $v_0(t,x)={\int }_{\Omega }u(t,x,y){\psi }_0\left(y\right)dy$, and based on this conclusion, the author further investigated the global asymptotic stability of the cylinder waves and the asymptotic behavior of the solution in the x direction as $t\to +\infty$ for nonnegative initial data decaying exponentially or vanishing as $\left|x\right|\to +\infty$. Suppose that ${\int }_{\Omega }{\psi }_0\left(y\right)K\left(y\right)dy=1-{\lambda }_0$, and denote $U_0\left(x\right):={\int }_{\Omega }{u}_0(x,y){\psi }_0\left(y\right)dy.$ When

$$\underset{x\to -\infty }{lim\; inf}{U}_0\left(x\right)>0,\underset{x\to \infty }{lim}{e}^{\sigma x}{U}_0\left(x\right)=r>0\mathrm{with}0<\sigma <\sqrt{1-{\lambda }_0},$$

it was shown that $\parallel u(t,z+ct,y)-V_c(z-{\sigma }^{-1}lnr){\psi }_0\left(y\right){\parallel }_{L_{\infty }(\mathbb{R}\times \Omega )}\to 0$ as $t\to \infty$. Moreover, the global exponential stability of cylinder waves in exponentially weighted space was also obtained when $U_0\left(x\right)\sim r{e}^{-\sigma x}+O\left(e^{-ax}\right)$ as $x\to +\infty$. When $u_0(x,y)$ is nonnegative and compactly supported in the cylinder, the global asymptotic stability of cylinder wave with critical speed was also studied in [3].

Note that (1) can be regarded as a simplified version of the nonlocal reaction-diffusion model studied in [5] as follows

$$\left\{\begin{array}{c}{\partial }_{t}u(t,x,y)-d_x{\Delta }_{x}u(t,x,y)-d_y{\Delta }_{y}u(t,x,y)\hfill \\ =\left[1-\alpha g(y-\theta )-{\int }_{\Omega }K(x,y,y^{\prime })u(t,x,y^{\prime })d{y}^{\prime }\right]u(t,x,y),\hfill & \hfill & t>0,x\in \mathbb{R},y\in \Omega ,\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}=0,\hfill & \hfill & t>0,x\in \mathbb{R},y\in \partial \Omega ,\hfill \\ u(0,x,y)=u_0(x,y),\hfill & \hfill & x\in \mathbb{R},y\in \Omega ,\hfill \end{array}\right.$$

(2)

where $\alpha$ stands for the intensity of selection towards the optimal value $\theta$, and $K(x,y,y^{\prime })$ implies that competitive ability depends not only on the spatial variable x but also on the traits $y,y^{\prime }$ of two competitive individuals. Note that (2) is reduced by $d_x=d_y=1$, $\theta =0$ and $K(x,y,y^{\prime })=K\left(y^{\prime }\right)$ in (1). There are many works on the cylinder front solution and the spreading speed of (2). We refer to [6] for the existence of a cylinder front with $\Omega ={\mathbb{R}}^n$ and $\theta \left(x\right)=bx$ and [7] for the existence of steady-state solutions and pulsating fronts with periodic $\theta \left(x\right)$, as well as [8] for the existence of cylinder waves and the spreading speed of (2) in the moving environment with $\theta \left(x\right)=b(x-ct)$. In addition, the spreading speed and the asymptotic behavior of the solution were studied in [9] with a bounded $\Omega$ and a constant kernel $K\equiv 1$ applying the Hamilton-Jacobi approach.

1.2. Main Assumptions and Results

In this paper, we consider the initial data decaying slower than any exponential function. Assume that there exists a function $w_0\left(x\right)$ such that

(H2) 

$C_1{w}_0\left(x\right){\psi }_0\left(y\right)⩽u_0(x,y)⩽C_2{w}_0\left(x\right){\psi }_0\left(y\right)$ for some constants $C_1,C_2>0$.

Denote

$$C_{\mathrm{unif}}^b\left(\mathbb{R}\right)=\{w\in C\left(\mathbb{R}\right)|w\mathrm{is}\mathrm{uniformly}\mathrm{continuous}\mathrm{in}x\in \mathbb{R}\mathrm{and}{sup}_{x\in \mathbb{R}}\left|w\left(x\right)\right|<\infty \},$$

which is equipped with the norm ${\parallel w\parallel }_{\infty }={sup}_{x\in \mathbb{R}}\left|w\left(x\right)\right|$. Throughout the paper, we assume that the initial function satisfies

$$w_0\in C_{\mathrm{unif}}^b\left(\mathbb{R}\right),w_0\left(x\right)>0\mathrm{for}x\in \mathbb{R},\underset{x\to -\infty }{lim\; inf}{w}_0\left(x\right)>0,\mathrm{and}{w}_0\left(x\right)\to 0\mathrm{as}x\to +\infty .$$

(3)

We say $w_0\left(x\right)$ is a slowly decaying function, if there is a large constant ${\xi }_0$ such that

$$w_0\in C^2\left([{\xi }_0,+\infty )\right),w_0^{\prime }⩽0\mathrm{in}[{\xi }_0,+\infty ),\mathrm{and}{w}_0^{\prime \prime }\left(x\right)/w_0\left(x\right)\to 0\mathrm{as}x\to +\infty .$$

(4)

Under (3) and (4), $w_0$ decays more slowly than any exponentially decaying function as $x\to +\infty$, that is,

$$\forall \kappa ,\exists x_{\kappa }s.t.w_0\left(x\right)⩾e^{-\kappa x}\mathrm{for}\mathrm{all}x\in [x_{\kappa },+\infty ).$$

Two typical examples of (3) and (4) are given by $w_0\left(x\right)=C{x}^{-p}$ for large x with $p,C>0$, or $w_0\left(x\right)=exp\{-x^q\}$ for large x with $q\in (0,1)$. Note that when $u_0(x,y)$ satisfies (H2), (3) and (4), we have the following.

$$C_1{w}_0\left(x\right)⩽U_0\left(x\right)={\int }_{\Omega }{u}_0(x,y){\psi }_0\left(y\right)dy⩽C_2{w}_0\left(x\right),$$

which indicates that the initial data considered in this paper are different from any of those studied in [3].

Therefore, we study the accelerating propagation of (1) with slowly decaying initial data, and the main theorem is stated as follows.

Theorem 1.

Let g and K satisfy (H1), and suppose that $u_0(x,y)$ satisfies (H2), (3) and (4). Then for any $y\in \Omega$, $\lambda \in (0,a/b)$, $\epsilon \in (0,a)$, $\gamma >0$ and $\Gamma >0$, there is $T=T_{y,\lambda ,\epsilon ,\gamma ,\Gamma }>0$ such that

$$E_y^{\lambda }\left(t\right)\subseteq w_0^{-1}(\gamma e^{-(a+\epsilon )t},\Gamma e^{-(a-\epsilon )t})\mathit{for}\mathit{any}t⩾T,$$

where $a=1-{\lambda }_0$, $b={\int }_{\Omega }K\left(y\right){\psi }_0\left(y\right)dy$, and $E_y^{\lambda }\left(t\right)=\{x\in \mathbb{R}|u(t,x,y)=\lambda {\psi }_0\left(y\right)\}$.

Example 1.

Under the same assumption in Theorem 1, we have that

(1)  if $w_0\left(x\right)\sim O\left(x^{-p}\right)$ as $x\to +\infty$, then

$$min{E}_y^{\lambda }\left(t\right)\sim max{E}_y^{\lambda }\left(t\right)\sim exp(at/p),t\to +\infty ,$$

(2)  if $u_0\left(x\right)\sim O\left(e^{-x^q}\right)$ with $q\in (0,1)$ as $x\to +\infty$, then

$$min{E}_y^{\lambda }\left(t\right)\sim max{E}_y^{\lambda }\left(t\right)\sim a^{{\displaystyle \frac{1}{q}}}{t}^{{\displaystyle \frac{1}{q}}},t\to +\infty .$$

Next, we consider the spatial propagation of (2) by applying the method in Theorem 1. The primary difficulty of this model stems from the complexity of the competition term ${\int }_{\Omega }K(x,y,y^{\prime })u(t,x,y^{\prime })d{y}^{\prime }$. For analytical convenience, we assume that

Description 1.

 

(H3) 

$g\in L_{\infty }(\Omega ),{\lambda }_0<1$, and there exist $D_1$ and $D_2$ such that

$$0<D_1\le {\int }_{\Omega }K(x,y,y^{\prime })d{y}^{\prime }\le D_2\mathrm{for}\mathrm{any}(x,y)\in \mathbb{R}\times \Omega .$$

Here ${\lambda }_0$ is the first eigenvalue of the operator $-d_y{\Delta }_y+\alpha g(y-\theta )$ under homogeneous Neumann boundary condition on $\partial \Omega$, and we also denote the corresponding positive eigenfunction by ${\psi }_0\left(y\right)$. Then it follows that

$$\left\{\begin{array}{cc}[-d_y{\Delta }_y+\alpha g(y-\theta )]{\psi }_0\left(y\right)={\lambda }_0{\psi }_0\left(y\right),\hfill & y\in \Omega ,\hfill \\ {\displaystyle \frac{\partial {\psi }_0\left(y\right)}{\partial \nu }}=0,\hfill & y\in \partial \Omega .\hfill \end{array}\right.$$

Our results for the acceleration propagation (2) are stated as follows.

Corollary 1.

Suppose that g and $K(x,y,y^{\prime })$ satisfy (H3), and let $u_0(x,y)$ satisfy (H2), (3) and (4). Then the model (2) has the same propagation properties as in Theorem 1.

Theorem 1 demonstrates that when the initial function exhibits slow decay characteristics (such as algebraic or sub-exponential decay), the solution of the nonlocal reaction-diffusion system (1) displays accelerating propagation. This means that for all subpopulations with distinct traits $y\in \Omega$, their respective invasion fronts propagate at ever-increasing speeds as time tends to infinity. It is commonly said that the solution has infinite speeding speed when it satisfies the propagation property in Theorem 1. This result indicates that in nonlocal reaction-diffusion models, the propagation speed of a population depends not only on the diffusion coefficient and local growth rate but also, and more profoundly, on the decay rate of the initial distribution in the far-field spatial region. Slowly decaying initial conditions provide a sustained “spatial momentum” to the population, driving an acceleration process that surpasses the classic constant-speed traveling wave propagation mode.

From model (1) to the more general model (2), this conclusion remains valid under the assumption (H3) on the competition kernel. Model (2) incorporates selection intensity $\alpha$, an optimal trait $\theta$, and a competition ability $K(x,y,y^{\prime })$ that depends on both space and traits, making it more general. Theorem 1 and Corollary 1 together illustrate that even with a more complex competition term, as long as the initial distribution decays slowly in the spatial direction and its trait structure can be separated from the spatial part via the eigenfunction ${\psi }_0\left(y\right)$, the essential feature of accelerating propagation—that level sets advance at an ever-increasing speed—persists. This reveals the profound dependence of propagation dynamics on the global geometric form of the initial condition (not merely its local support) in nonlocal models with spatial and trait structures.

From a mathematical perspective, “infinite spreading speed” means that the velocity of the propagation front grows without bound as time tends to infinity, rather than representing instantaneous transmission in the physical sense. In ecological contexts, this corresponds to species dispersing over very long distances in relatively short time scales—such as plant seeds being carried by wind, birds, or human activities that facilitate long-distance dispersal events. If we interpret the variable x differently—for example, by viewing it as the latitude on Earth—then the phenomenon of accelerating propagation in the model could be understood as the rapid expansion of a certain type of bird or insect, adapted to various latitudes, spreading from the equator (or a specific region) toward the North and South Poles.

Ecologically, this finding carries significant implications. It suggests that when a species (or a specific trait) exhibits a “long-tailed” distribution at the periphery of the introduced region—where density is very low but decays extremely slowly—its subsequent spatial expansion may far exceed predictions based on local diffusion or exponentially decaying initial conditions. For instance, in conservation biology and invasion ecology, assessing the potential expansion risk of a species solely by focusing on the density of its core distribution area or assuming rapid decay at its boundaries may severely underestimate its long-term dispersal capacity. This is particularly relevant for species capable of long-distance dispersal (e.g., via wind, water, or animal vectors), as their initial distributions are more likely to exhibit such slow decay characteristics, leading to unexpectedly rapid range expansion. Furthermore, in scenarios such as the intentional release of gene-drive organisms or genetically modified organisms, fine control over the spatial distribution of initially introduced individuals—especially avoiding the creation of low-density “pioneer” individuals at long distances—may be crucial for managing their propagation speed. This study provides a theoretical framework for quantitatively assessing propagation potential in such contexts.

1.3. Research Motivation for Slowly Decaying Initial Data and Its Induced Accelerating Propagation

The slowly decaying initial conditions—such as algebraic or sub-exponential decay—considered in this study are ecologically and biologically well-motivated. In many real-world scenarios of population dispersal and biological invasion, the initial spatial distribution of a species is often not compactly supported or exponentially bounded. Some examples are listed as follows.

Long-distance dispersal events: Seeds, insects, or microorganisms can be transported over long distances by wind, water currents, birds, or airplanes, resulting in sparse “pioneer individuals” far from the core population. Their density may be extremely low but decays slowly in space rather than abruptly vanishing.

Non-uniform introductions or historical remnants: In cases of intentional species release (e.g., biocontrol agents and genetically modified organisms) or residual populations after range contraction, the initial distribution may exhibit a “long-tail” pattern—very low density extending over a wide spatial domain.

Environmental heterogeneity and diverse dispersal strategies: In heterogeneous environments, populations may adopt multiple dispersal strategies, with some individuals moving to distant areas to explore new resources, thereby generating a slowly decaying spatial profile.

Therefore, studying propagation dynamics under slowly decaying initial data is crucial for accurately assessing the invasion risk of species that are capable of long-distance dispersal or that initially exhibit widespread but sparse distributions (e.g., invasive species, disease hosts, and genetically engineered organisms). The theoretical analysis presented here provides a mathematical foundation for understanding the phenomenon of accelerating propagation in such realistic contexts.

Slowly decaying initial data are not only pivotal in this study but also have extensive practical correspondence in numerous partial differential equations and ecological models, garnering sustained attention. In the classical reaction-diffusion equation

$$\left\{\begin{array}{ccc}{u}_t=u_{xx}+f\left(u\right),\hfill & \hfill & t>0,x\in \mathbb{R},\hfill \\ u(0,x)=u_0\left(x\right),\hfill & \hfill & x\in \mathbb{R},\hfill \end{array}\right.$$

(5)

Hamel and Roques [10] showed that the solution of (5) has an infinite spreading speed under the Fisher-KPP assumption on f, in the sense that, the level sets of the solutions move infinitely fast as time goes to infinity, and more accurate descriptions on the locations of level sets were also obtained. Subsequently, the accelerating propagation of (5) with degenerated reaction function (namely $f^{\prime }\left(0\right)=0$) was studied in [11] and the exact separation between no acceleration and acceleration was provided. We refer to [12] for the results about accelerating propagation in nonlocal dispersal equations. It is worth pointing out that the accelerating invasion was also studied in [13] for the nonlocal reaction-diffusion model, where the optimal trait for survival depends linearly on the spatial variable, as follows

$$\left\{\begin{array}{cc}{\partial }_{t}u-u_{xx}-u_{yy}=\left[r(y-Bx)-{\int }_{\mathbb{R}}K(t,x,y,y^{\prime })u(t,x,y^{\prime })d{y}^{\prime }\right]u,\hfill & t>0,(x,y)\in {\mathbb{R}}^2,\hfill \\ u(0,x,y)=u_0(x,y),\hfill & (x,y)\in {\mathbb{R}}^2,\hfill \end{array}\right.$$

when the initial data satisfy the heavy-tailed condition in the direction $y-bx=0$ on the space-trait plane. Moreover, the accelerating propagation of the following reaction-diffusion equation in cylinder

$$\left\{\begin{array}{cc}{u}_t-u_{xx}-{\Delta }_{y}u=f(y,u)\hfill & t>0,x\in \mathbb{R},y\in \Omega ,\hfill \\ u(0,x,y)=u_0(x,y)\hfill & x\in \mathbb{R},y\in \Omega .\hfill \end{array}\right.$$

was studied in [14], when $u_0(·,y)$ is a slowly decaying function for some $y\in \mathbb{R}$.

The remainder of this paper is organized as follows. Section 2 proves the main results (Theorem 1) on accelerating propagation for model (1). Section 3 extends the key conclusions to the more general model (2) and prove the corresponding corollary (Corollary 1). Finally, Section 4 provides a summary and discussion of the mathematical implications and ecological insights derived from the obtained results.

2. The Proof of Theorem 1

Consider two functions $\underline{u}(t,x,y)=w_1(t,x){\psi }_0\left(y\right)$ and $\overline{u}(t,x,y)=w_2(t,x){\psi }_0\left(y\right)$, where $w_i(t,x)$ satisfies

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\partial w_i}{\partial t}}-{\displaystyle \frac{{\partial }^2{w}_i}{\partial x^2}}=\left[1-{\lambda }_0-w_i{\int }_{\Omega }K\left(y\right){\psi }_0\left(y\right)dy\right]w_i,\hfill & \hfill & t>0,x\in \mathbb{R},\hfill \\ w_i(0,x)=C_i{w}_0\left(x\right),\hfill & \hfill & x\in \mathbb{R}.\hfill \end{array}\right.$$

(6)

We can verify that $\overline{u}(t,x,y)$ and $\underline{u}(t,x,y)$ are a pair of super- and sub-solutions of (1). Then the comparison principle of (1) implies that

$$w_1(t,x){\psi }_0\left(y\right)⩽u(t,x,y)⩽w_2(t,x){\psi }_0\left(y\right),t>0,x\in \mathbb{R},y\in \Omega .$$

(7)

In what follows, we study the properties of $w_1(t,x)$ and $w_2(t,x)$.

Upper bounds for level sets. In this part, we consider $w_2(t,x)$. By (4), there exists a constant ${\xi }_1>{\xi }_0$ such that

$$|w_0^{″}\left(x\right)|⩽{\displaystyle \frac{\epsilon }{2}}{w}_0\left(x\right)\mathrm{for}\mathrm{any}x⩾{\xi }_1.$$

Recall that $a=1-{\lambda }_0$ and $b={\int }_{\Omega }K\left(y\right){\psi }_0\left(y\right)dy$. We define a function $\theta (·):[0,+\infty )\to (0,+\infty )$ satisfying

$$\left\{\begin{array}{c}{\theta }^{\prime }\left(t\right)=\theta \left(t\right)(a-b\theta \left(t\right)),t>0,\hfill \\ \theta \left(0\right)={\theta }_0,\hfill \end{array}\right.$$

where ${\theta }_0=max\left\{{sup}_{x\in \mathbb{R}}\left\{C_2{w}_0\left(x\right)\right\},a/b\right\}$. Note that $\theta (·)$ is nonincreasing. We also have that if ${sup}_{x\in \mathbb{R}}\left\{C_2{w}_0\left(x\right)\right\}⩽a/b$, then $\theta \left(t\right)\equiv a/b$. Define

$$\overline{w}(t,x)=min\left\{D{w}_0\left(x\right)e^{\rho t},\theta \left(t\right)\right\}\mathrm{for}x\in \mathbb{R},t⩾0,$$

(8)

where

$$\rho =a+{\displaystyle \frac{\epsilon }{2}},D={\displaystyle \frac{{\theta }_0}{w_0\left({\xi }_1\right)}}.$$

When $\overline{u}(t,x)=D{w}_0\left(x\right)e^{\rho t}<\theta \left(t\right)$, we have that $x⩾{\xi }_1$ and

$$\begin{array}{cc}\hfill \mathcal{L}\overline{w}:=& {\overline{w}}_t-{\overline{w}}_{xx}-\left[1-{\lambda }_0-\overline{w}{\int }_{\Omega }K\left(y\right){\psi }_0\left(y\right)dy\right]\overline{w}\hfill \\ \hfill ⩾& D{e}^{\rho t}[\rho w_0\left(x\right)-w_0^{\prime \prime }\left(x\right)-a{w}_0\left(x\right)]\hfill \\ \hfill =& D{e}^{\rho t}\left[{\displaystyle \frac{\epsilon }{2}}{w}_0\left(x\right)-w_0^{\prime \prime }\left(x\right)\right]⩾0.\hfill \end{array}$$

When $\overline{u}(t,x)=\theta \left(t\right)$, we can easily check that $\mathcal{L}\overline{w}⩾0$. Note that

$$D={\displaystyle \frac{{\theta }_0}{w_0\left({\xi }_1\right)}}⩾{\displaystyle \frac{{sup}_{x\in \mathbb{R}}\left\{C_2{w}_0\left(x\right)\right\}}{w_0\left({\xi }_1\right)}}⩾C_2.$$

When $t=0$, it holds that

$$\overline{w}(0,x)=min\left\{D{w}_0\left(x\right),{\theta }_0\right\}⩾C_2{w}_0\left(x\right)=w_2(0,x)\mathrm{for}x\in \mathbb{R}.$$

The comparison principle of (6) implies that

$$w_2(t,x)⩽\overline{w}(t,x),t>0,x\in \mathbb{R}.$$

By (7), we have that

$$u(t,x,y)⩽w_2(t,x){\psi }_0\left(y\right)⩽min\left\{D{w}_0\left(x\right)e^{\rho t}{\psi }_0\left(y\right),\theta \left(t\right){\psi }_0\left(y\right)\right\},t>0,x\in \mathbb{R},y\in \Omega .$$

For any $t>0$, $y\in \Omega$ and $\lambda \in (0,a/b)$, let $x_y^{\lambda }\left(t\right)$ be an arbitrary element in $E_y^{\lambda }\left(t\right)$. Then

$$\lambda {\psi }_0\left(y\right)=u(t,x_y^{\lambda }\left(t\right),y)⩽D{w}_0\left(x_y^{\lambda }\left(t\right)\right)e^{\rho t}{\psi }_0\left(y\right).$$

Therefore, for any $\gamma >0$, there is a sufficiently large $T_1>0$ such that

$$w_0\left(x_y^{\lambda }\left(t\right)\right)⩾\lambda D^{-1}{e}^{-\rho t}>\gamma e^{-(a+\epsilon )t}\mathrm{for}\mathrm{any}t>T_1.$$

which implies that

$$x_y^{\lambda }\left(t\right)\in w_0^{-1}\left(\gamma e^{-(a+\epsilon )t},+\infty \right)\mathrm{for}\mathrm{any}t>T_1.$$

(9)

Lower bounds for level sets. Next, we consider $w_1(t,x)$. For any $\epsilon \in (0,a)$, let $\rho >0$ satisfy that

$$max\{a-\epsilon ,a/2\}<\rho <a.$$

By (3) and (4), it follows from Lemma 2.2 in [15] that $w_0^{\prime }\left(x\right)/w_0\left(x\right)\to 0$ as $x\to +\infty$. Then we can choose ${\xi }_2⩾{\xi }_0$ such that

$$\begin{array}{c}{\left(w_0^{\prime }\left(x\right)\right)}^2⩽l_1{\left(w_0\left(x\right)\right)}^2,\left|w_0^{″}\left(x\right)\right|⩽l_2{w}_0\left(x\right)\mathrm{for}\mathrm{any}x⩾{\xi }_2.\hfill \end{array}$$

(10)

$$\begin{array}{c}{w}_0\left(x\right)>\kappa :=w_0\left({\xi }_2\right)\mathrm{for}\mathrm{any}x<{\xi }_2,\hfill \end{array}$$

(11)

where

$$l_1={\displaystyle \frac{2\rho -a}{8}},l_2=min\left\{a-\rho ,{\displaystyle \frac{2\rho -a}{8}}\right\}.$$

(12)

Define $g\left(s\right)=s-A{s}^2$ for $s>0$, where

$$A=max\left(C_1^{-1}{\kappa }^{-1},{\displaystyle \frac{2b}{2\rho -a}}\right).$$

(13)

Then

$$g\left(s\right)>0\mathrm{for}s\in (0,1/A),g\left(s\right)<0\mathrm{for}s\in (1/A,+\infty ),$$

(14)

and

$$\underset{s>0}{max}\left\{g\left(s\right)\right\}=g\left({\displaystyle \frac{1}{2A}}\right)={\displaystyle \frac{1}{4A}}⩽{\displaystyle \frac{C_1\kappa }{4}}.$$

(15)

Define

$$\underline{w}(t,x)=max\left\{0,g\left(C_1{w}_0\left(x\right)e^{\rho t}\right)\right\}=max\left\{0,C_1{w}_0\left(x\right)e^{\rho t}-A{C}_1^2{w}_0^2\left(x\right)e^{2\rho t}\right\}.$$

(16)

When $\underline{w}(t,x)=g\left(C_1{w}_0\left(x\right)e^{\rho t}\right)>0$, we get from (13) and (14) that $C_1{w}_0\left(x\right)e^{\rho t}<1/A<C_1\kappa$, which along with (11) implies that $x⩾{\xi }_2$. Then it follows that

$$\begin{array}{cc}\hfill \mathcal{L}\underline{w}⩽& \rho C_1{w}_0\left(x\right)e^{\rho t}-2\rho A{C}_1^2{w}_0^2\left(x\right)e^{2\rho t}\hfill \\ \hfill & -C_1{w}_0^{″}\left(x\right)e^{\rho t}+2A{C}_1^2{\left(w_0^{\prime }\left(x\right)\right)}^2{e}^{2\rho t}+2A{C}_1^2{w}_0\left(x\right)w_0^{″}\left(x\right)e^{2\rho t}\hfill \\ \hfill & -a{C}_1{w}_0\left(x\right)e^{\rho t}+aA{C}_1^2{w}_0^2\left(x\right)e^{2\rho t}+b{C}_1^2{w}_0^2\left(x\right)e^{2\rho t}\hfill \\ \hfill =& C_1{e}^{\rho t}{H}_1\left(x\right)+C_1^2{e}^{2\rho t}{H}_2\left(x\right),\hfill \end{array}$$

where

$$\begin{array}{c}{H}_1\left(x\right):=\rho w_0\left(x\right)-w_0^{″}\left(x\right)-a{w}_0\left(x\right),\hfill \\ H_2\left(x\right):=A(-2\rho w_0^2\left(x\right)+2{\left(w_0^{\prime }\left(x\right)\right)}^2+2w_0\left(x\right)w_0^{″}\left(x\right)+a{w}_0^2\left(x\right))+b{w}_0^2\left(x\right).\hfill \end{array}$$

It follows from (10) and (12) that

$$H_1\left(x\right)⩽w_0\left(x\right)(\rho +l_2-a)⩽0,$$

and along with (13), we get that

$$\begin{array}{cc}\hfill H_2\left(x\right)⩽& w_0^2\left(x\right)[A(-2\rho +2l_1+2l_2+a)+b]⩽0.\hfill \end{array}$$

Then $\mathcal{L}\underline{w}(t,x)⩽0$ holds. When $\underline{w}(t,x)=0$, we easily check that $\mathcal{L}\underline{w}(t,x)⩽0$. It follows that $\mathcal{L}\underline{w}(t,x)⩽0$ for any $t>0$ and $x\in \mathbb{R}$. The spatial homogeneity of $\mathcal{L}$ implies that

$$\mathcal{L}\underline{w}(t,x+X)⩽0\mathrm{for}\mathrm{any}t>0,x\in \mathbb{R},X⩾0.$$

By (11), (15), (16) and the monotonicity of $w_0$ on $[{\xi }_0,+\infty )$, we have that

$$\underline{w}(0,x+X)⩽min\{C_1{w}_0(x+X),C_1\kappa \}⩽C_1{w}_0\left(x\right)=w_1(0,x)\mathrm{for}x\in \mathbb{R},X⩾0.$$

Then the comparison principle of (6) implies that

$$\underline{w}(t,x+X)⩽w_1(t,x)\mathrm{for}\mathrm{any}t>0,x\in \mathbb{R},X⩾0.$$

(17)

Let $x_M\left(t\right)$ be the function satisfying $w_0\left(x_M\left(t\right)\right)={\left(2A{C}_1\right)}^{-1}{e}^{-\rho t}$, which means $\underline{w}(t,x_M\left(t\right))=g\left({\displaystyle \frac{1}{2A}}\right)={\left(4A\right)}^{-1}$. It is easy to check that $x_M\left(t\right)⩾{\xi }_2$. By (17), we have that

$$w_1(t,x_M\left(t\right)-X)⩾\underline{w}(t,x_M\left(t\right))={\left(4A\right)}^{-1}\mathrm{for}\mathrm{any}t>0,X⩾0,$$

which means

$$w_1(t,x)⩾{\left(4A\right)}^{-1}\mathrm{for}\mathrm{any}t>0,x⩽x_M\left(t\right).$$

By (7), we have that

$$u(t,x,y)⩾w_1(t,x){\psi }_0\left(y\right)⩾{\left(4A\right)}^{-1}{\psi }_0\left(y\right)\mathrm{for}\mathrm{any}t>0,x⩽x_M\left(t\right),y\in \Omega .$$

For any $\lambda \in (0,{\left(4A\right)}^{-1})$ and any $x_y^{\lambda }\left(t\right)\in E_y^{\lambda }\left(t\right)$, we have that

$$u(t,x_y^{\lambda }\left(t\right),y)=\lambda {\psi }_0\left(y\right)<{\left(4A\right)}^{-1}{\psi }_0\left(y\right),$$

which implies that $x_y^{\lambda }\left(t\right)>x_M\left(t\right)⩾{\xi }_2$. For any $\Gamma >0$, there is $T_2>0$ large enough such that

$$w_0\left(x_y^{\lambda }\left(t\right)\right)⩽w_0\left(x_M\left(t\right)\right)={\left(2A{C}_1\right)}^{-1}{e}^{-\rho t}<\Gamma e^{-(a-\epsilon )t}\mathrm{for}t⩾T_2.$$

It follows that

$$x_y^{\lambda }\left(t\right)\in w_0^{-1}(0,\Gamma e^{-(a-\epsilon )t})\mathrm{for}t⩾T_2.$$

Now consider the general case $\lambda \in (0,a/b)$. By a similar method to the proof of (3.29) in [15], for any $\Gamma >0$, we can get that

$$\underset{t\to +\infty }{lim}\underset{x⩽\eta \left(t\right)}{sup}\left|w_1(x,t)-a/b\right|=0,$$

where $\eta \left(t\right)$ is an arbitrary function satisfying $w_0\left(\eta \left(t\right)\right)=\Gamma e^{-(a-ϵ)t}$ for t large enough. Then for any $\lambda \in (0,a/b)$, there exists $T_2^{\prime }⩾0$ such that

$$\underset{x⩽\eta \left(t\right)}{inf}{w}_1(x,t)>\lambda \mathrm{for}\mathrm{any}t⩾T_2^{\prime }.$$

(18)

For any $x_y^{\lambda }\left(t\right)\in E_y^{\lambda }\left(t\right)$, we have that

$$\lambda {\psi }_0\left(y\right)=u(t,x_y^{\lambda }\left(t\right),y)⩾w_1(t,x_y^{\lambda }\left(t\right)){\psi }_0\left(y\right),$$

which implies that $w_1(t,x_y^{\lambda }\left(t\right))⩽\lambda$. By (18), we have that $x_y^{\lambda }\left(t\right)>\eta \left(t\right)$ for $t⩾T_2^{\prime }$. The arbitrariness of $\eta \left(t\right)$ shows that for any $\lambda \in (0,a/b)$,

$$x_y^{\lambda }\left(t\right)\in w_0^{-1}(0,\Gamma e^{-(a-\epsilon )t})\mathrm{for}t⩾T_2^{\prime }.$$

(19)

is the eigenvector corresponding to the principal eigenvalue Take $T=max\{T_1,T_2^{\prime }\}$, and the proof of Theorem 1 is completed by (9) and (19).is the eigenvector corresponding to the principal eigenvalue.

3. The Proof of Corollary 1

Note that ${\psi }_0$ is the eigenfunction corresponding to the first eigenvalue ${\lambda }_1$ of the operator $-d_y{\Delta }_y+\alpha g(y-\theta )$ with Neumann boundary condition. Then there exist two constant $B_1$ and $B_2$ such that

$$0<B_1\le {\psi }_0\left(y^{\prime }\right)\le B_2\mathrm{for}\mathrm{any}{y}^{\prime }\in \Omega .$$

By (H3), it holds that

$$0<B_1{D}_1\le {\int }_{\Omega }K(x,y,y^{\prime }){\psi }_0\left(y^{\prime }\right)d{y}^{\prime }\le B_2{D}_2\mathrm{for}\mathrm{any}(x,y)\in \mathbb{R}\times \Omega .$$

(20)

Consider two functions $\underline{u}(t,x,y)=w_1(t,x){\psi }_0\left(y\right)$ and $\overline{u}(t,x,y)=w_2(t,x){\psi }_0\left(y\right)$, where $w_i(t,x)$ satisfies

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\partial w_i}{\partial t}}-{\displaystyle \frac{{\partial }^2{w}_i}{\partial x^2}}=\left[1-{\lambda }_0-B_i{D}_i{w}_i\right]w_i,\hfill & \hfill & t>0,x\in \mathbb{R},\hfill \\ w_i(0,x)=C_i{w}_0\left(x\right),\hfill & \hfill & x\in \mathbb{R}.\hfill \end{array}\right.$$

(21)

By (20), we see that ${\underline{u}}_1$ is a lower solution and ${\overline{u}}_2$ is an upper solution of (2). Since the level sets of the solution for (21) have the same upper and lower bound estimations as discussed in Section 2, the solution to (2) has the same propagation properties as in Theorem 1.

4. Discussion and Conclusions

This paper investigates the phenomenon of accelerating propagation in nonlocal reaction-diffusion models with spatial and trait structures under slowly decaying initial conditions. We established upper and lower bounds for the propagation of level sets for two models—the basic model (1) and the more general model (2)—and elucidated the fundamental characteristics of their accelerating propagation.

4.1. Summary of Main Findings

Theorem 1 demonstrates that for the basic model (1), as long as the initial data satisfy condition (H2) and its spatial decay is slower than any exponential function (e.g., algebraic decay or sub-exponential decay), the propagation speed of the solution tends to infinity, i.e., accelerating propagation occurs. Specifically, for any fixed density level $\lambda$, the position of its corresponding spatial level set $E_y^{\lambda }\left(t\right)$ advances at an ever-increasing speed, thereby proving that the invasion front advances in a continuously accelerating manner.

Corollary 1 extends the above conclusion to the more general model (2). Despite the introduction of selection intensity $\alpha$, an optimal trait $\theta$, and a competition kernel $K(x,y,y^{\prime })$ dependent on both space and traits, under assumption (H3), as long as the spatial decay of the initial distribution is sufficiently slow and its trait structure can be separated from the spatial part via the principal eigenfunction ${\psi }_0$, the essential feature of accelerating propagation—namely, the level sets advancing at an ever-increasing speed—remains intact.

These results collectively reveal a profound mathematical insight: in nonlocal models with spatial and trait structures, the propagation dynamics of a population depend not only on the diffusion coefficient and local growth rate but, more fundamentally, on the decay rate of the initial distribution in the far-field spatial region. Slowly decaying initial conditions provide the population with sustained “spatial momentum,” driving it beyond the classic constant-speed traveling wave propagation mode into a state of accelerating propagation.

4.2. Connections to Numerical Simulation and Optimal Control

While this work provides a rigorous theoretical analysis of accelerating propagation under slowly decaying initial data, our results naturally point toward two important applied directions: numerical validation and controlled propagation. From a computational perspective, the accelerating fronts predicted by Theorem 1 pose a significant challenge for numerical simulation, due to the nonlocal coupling in the reaction term and the exponentially growing propagation speed. Recent advances in the numerical treatment of nonlocal continuity equations—such as those developed by Chertovskih et al. [16] using measure-theoretic optimal control and Pontryagin-type maximum principles—offer promising frameworks for simulating such dynamics. Their descent-based algorithms, designed for nonlinear nonlocal transport equations in probability measure spaces, could be adapted to numerically verify the accelerating propagation phenomena established here, providing a valuable bridge between theory and computation.

Moreover, from an applied ecological standpoint, understanding how a population spreads is often a precursor to asking whether and how one might control its spread—for instance, to mitigate biological invasions or to design spatial conservation strategies. The same work by Chertovskih et al. directly links nonlocal dynamics with optimal control theory, offering a methodological pathway to extend our results to controlled settings. By citing this emerging line of research, we not only contextualize our theoretical findings within the broader landscape of nonlocal PDE analysis, but also highlight their potential relevance for real-world problems in population management, invasive species control, and spatially structured ecological planning.

4.3. Theoretical Significance and Ecological Implications

From a theoretical perspective, this study expands our understanding of propagation mechanisms in nonlocal models. It indicates that even with identical local dynamics and forms of nonlocal competition, the global geometric form of the initial conditions (rather than merely their local support) can fundamentally alter the long-term spatiotemporal patterns of population invasion. In terms of ecological applications, this research offers the following important implications.

(1) Risk assessment: In conservation biology and invasion ecology, when assessing species expansion risks, focusing solely on the density of the core distribution area or assuming rapid decay at its boundaries may severely underestimate its long-term dispersal potential. This is particularly relevant for species capable of long-distance dispersal (e.g., via wind, water, or animals), as their initial distributions are more likely to exhibit slow decay characteristics, leading to actual expansion speeds far exceeding predictions based on exponential decay assumptions.

(2) Biotechnology management: In scenarios such as the intentional release of gene-drive organisms or genetically modified organisms, fine control over the spatial distribution of initially introduced individuals is crucial. Special care should be taken to avoid the creation of low-density “pioneer” individuals at long distances to prevent their accelerated propagation. This study provides a theoretical framework for quantitatively assessing propagation potential in such contexts.

4.4. Limitations and Future Directions

The primary limitation of this study lies in the structural assumption (H2) required for the initial conditions, i.e., the spatial profile $u_0(·,y)$ of the initial distribution must be controlled (up to a positive constant multiple) by a single slowly decaying function $w_0\left(x\right)$ for all traits $y\in \Omega$. This ensures that the propagation process is uniform with respect to the trait $y\in \Omega$.

However, in many practical situations, the initial distribution may not have such uniformity. For example, when genetically modified individuals are introduced into a region to improve the genetic characteristics of a population, their initial distribution may have compact support for some traits while exhibiting slow (or exponential) decay for others. That is, $\Omega$ can be decomposed into ${\Omega }_1$ and ${\Omega }_2$, so that $u(·,y)$ has compact support for $y\in {\Omega }_1$, while $u(·,y)$ is a slowly decaying function for $y\in {\Omega }_2$. Under such conditions where (H2) is not satisfied, the propagation behavior of the solution becomes more intricate, and the propagation front may exhibit trait-dependent, non-uniform advancement patterns. This represents an important direction for future research.