  In this article, we prove existence and uniqueness of the Barenblatt solution
of the evolution equation on the whole Euclidean space where the principle part
is the nonlocal fractional p-Laplacian composed with a power function. Our
proof generalizes methods developped by J.-L. Vazquez [Nonlinear Anal., 199
(2022), Calc. Var. Partial Differential Equations, 60 (2021)] for the evolution
equation driven by the fractional p-Laplacian on the whole Euclidean space. In
particular, we required an Aleksandrov symmetry principle, which can be applied
to the mild solutions of the evolution equation in $L^1$ governed by the doubly
nonlinear nonlocal operator, and the construction of global barrier functions.
The Aleksandrov symmetry principle might be of independent interest.
