Doubly nonlocal Fisher-KPP equation: Front propagation

We study propagation over $\mathbb{R}^d$ of the solution to a nonlocal nonlinear equation with anisotropic kernels, which can be interpretted as a doubly nonlocal reaction-diffusion equation of the Fisher--KPP-type. We prove that if the kernel of the nonlocal diffusion is exponentially integrable in a direction and if the initial condition decays in this direction faster than any exponential function, then the solution propagates at most linearly in time in that direction. Moreover, if both the kernel and the initial condition have the above properties in any direction (being, in general, anisotropic), then we prove linear in time propagation of the corresponding solution over $\mathbb{R}^d$.