Growing solutions of the fractional p-Laplacian equation in the Fast Diffusion range

We establish existence, uniqueness as well as quantitative estimates for solutions u(t, x) to the fractional nonlinear diffusion equation, partial derivative(t)u + L-s,L-p(u) = 0, where L-s,L-p = (-Delta)(p)(s) is the standard fractional p-Laplacian operator. We work in the range of exponents 0 < s < 1 and 1 < p < 2, and in some sections we need sp < 1. The equation is posed in the whole space x is an element of R-N. We first obtain weighted global integral estimates that allow establishing the existence of solutions for a class of large data that is proved to be roughly optimal. We use the estimates to study the class of self-similar solutions of forward type, that we describe in detail when they exist. We also explain what happens when possible self-similar solutions do not exist. We establish the dichotomy positivity versus extinction for nonnegative solutions at any given time. We analyse the conditions for extinction in finite time. (C) 2021 The Author (s). Published by Elsevier Ltd.