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

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.