Sharp exponent of acceleration in general nonlocal equations with a weak Allee effect

TL;DR

This work analyzes propagation in nonlocal, monostable equations with a weak Allee effect, proving a sharp algebraic acceleration rate for level-set propagation under a broad class of dispersal kernels that includes the fractional Laplacian and convolution-type operators. The authors develop tail-flattening estimates and a novel subsolution construction that captures near-front nonlinear growth together with far-field dissipative spreading, enabling a matching lower bound to known upper bounds. Under the condition $(2s-1)(\beta-1)<1$, they establish $x_\lambda(t) \asymp_\lambda t^{\beta/(2s(\beta-1))}$, revealing a unified acceleration regime across kernel types and nonlinearities. A numerical study in the fractional case corroborates the predicted scaling and highlights regime transitions as $s$ and $\beta$ vary, underscoring the practical relevance for ecological dispersal with heavy-tailed jumps.

Abstract

We study an acceleration phenomenon arising in monostable integro-differential equations with a weak Allee effect. Previous works have shown its occurrence and have given correct upper bounds on the rate of expansion in some particular cases, but precise lower bounds were still missing. In this paper, we provide a sharp lower bound for this acceleration rate, valid for a large class of dispersion operators. Our results manage to cover fractional Laplace operators and standard convolutions in a unified way, which is new in the literature. A first very important result of the paper is a general flattening estimate of independent interest: this phenomenon appears regularly in acceleration situations, but getting quantitative estimates is most of the time open. This estimate at hand, we construct a very subtle sub-solution that captures the expected dynamics of the accelerating solution (rates of expansion and flattening) and identifies several various regimes that appear in the dynamics depending on the parameters of the problem.

Figures (11)

• Figure 1: Subfigure (a): summary of the existing results for the convolution case. In the blue zone ➊, only an upper bound has been derived Alfaro2017: $x_\lambda(t) \lesssim_\lambda t^{\frac{\beta}{2s(\beta-1)}}$. In the green zone ➋, the model enjoys linear propagation with existence of travelling fronts Coville2006: $x_\lambda(t) \asymp t$. In the white zone ➌, no estimates are known. In the violet zone ➍, nonmatching lower and upper bounds have been obtained Alfaro2017: $t^{\frac{1}{2s(\beta-1)} } \lesssim_\lambda x_{\lambda}(t) \lesssim_\lambda t^{\frac{\beta}{2s(\beta-1)} }$. In the orange zone, exponential propagation occurs Garnier2011Bouin2018: $x_\lambda(t)\asymp_\lambda\exp({\rho} t)$. Subfigure (b): summary of the existing results for the fractional case. In the blue zone ➊, an upper bound was derived in Coville2021, with a matching lower bound found by the authors in an early version of the present work Bouin2021-hal and, independently, by Zhang and Zlatoš in Zhang2023: $x_\lambda(t) \asymp_\lambda t^{\frac{\beta}{2s(\beta-1)}}$. In the green zone ➋, the model enjoys linear propagation with existence of travelling fronts Gui2015Coville2021: $x_\lambda(t) \asymp t$. In the orange zone, exponential propagation occurs Cabre2013: $x_\lambda(t)\asymp_\lambda\exp( \rho t)$.
• Figure 2: In the blue zone ➊, sharp lower and upper bounds are provided by \ref{['thm:main']}: $x_\lambda(t)\asymp_\lambda t^{\frac{\beta}{2s(\beta-1)}}$. In the green zone ➋, the model enjoys a linear propagation with existence of travelling fronts Alfaro2017Coville2006Coville2021Gui2015: $x_\lambda(t)\asymp t$. In the orange zone, exponential propagation occurs, by straightforward extension of the work of Bouin et al.Bouin2018: $x_\lambda(t) \asymp_\lambda \exp(\rho t)$.
• Figure 3: Evolution over time of the constant $D$ obtained by fitting the function $\frac{D}{x^{2s}}$ with the part of the tail between values $10^{-2}$ and $10^{-5}$ of the numerical approximation of the solution to the fractional diffusion-reaction problem, with $\beta=1.5$ and $s=0.4$ (left) or $0.5$ (right).
• Figure 4: Schematic view of the expected behaviour of solution at a given time $t$. The solution travels at speed $t^{\frac{\beta}{2s(\beta-1)}}$ and the bulk is $t^{\frac{1}{2s}}$ wide.
• Figure 5: Schematic view of the subsolution $\underline{u}$ at a given time $t$. For estimations, several zones are considered. The blue zone is where $\underline{u}$ is constant, making computations easier. In the orange zone, we use crucially the fact that $\underline{u}$ looks like a solution to an ordinary differential equation of the form $n'=n^\beta$. The exact expression of $Y(t)$ will appear naturally later. In the brown (far-field) zone, a decay imitating that of the solution to a fractional Laplace equation provides the correct asymptotic behaviour. Finally, the construction in the intermediate green zone is more subtle and based on a mixture of the treatments in both surrounding zones.

Theorems & Definitions (46)

• Theorem 1.5
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• Proposition 2.4
• proof
• Claim 2.5
• proof : Proof of Claim \ref{['claim subsolution']}