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Doubly nonlocal Fisher–KPP equation: front propagation

ABSTRACT We study propagation over of the solution to a doubly nonlocal reaction–diffusion equation of the Fisher–KPP-type with anisotropic kernels. We present both necessary and sufficient conditions which ensure linear in time propagation of the solution in a direction. For kernels with a finite exponential moment over we prove front propagation in all directions for a general class of initial conditions decaying in all directions faster than any exponential function (that includes, for the first time in the literature about the considered type of equations, compactly supported initial conditions).