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Accelerating solutions in integro-differential equations

In this paper, we study the spreading properties of the solutions of an integro-differential equation of the form $u_t=J\ast u-u+f(u).$ We focus on equations with slowly decaying dispersal kernels $J(x)$ which correspond to models of population dynamics with long-distance dispersal events. We prove that for kernels $J$ which decrease to $0$ slower than any exponentially decaying function, the level sets of the solution $u$ propagate with an infinite asymptotic speed. Moreover, we obtain lower and upper bounds for the position of any level set of $u.$ These bounds allow us to estimate how the solution accelerates, depending on the kernel $J$: the slower the kernel decays, the faster the level sets propagate. Our results are in sharp contrast with most results on this type of equation, where the dispersal kernels are generally assumed to decrease exponentially fast, leading to finite propagation speeds.