Sharp exponent of acceleration in general non-local equations with a weak Allee effect

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, which is 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. An 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 in general an open question. With this estimate at hand, we construct a subsolution that captures the expected behaviour of the accelerating solution (rates of expansion and flattening) and identifies several regimes that appear in the dynamics depending on the parameters of the problem.