The Barenblatt solution of an evolution problem governed by a doubly nonlinear nonlocal operator

Abstract: 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 $L<sup>1$ governed by the doubly nonlinear nonlocal operator, and the construction of global barrier functions. The Aleksandrov symmetry principle might be of independent interest.