Growing Solutions of the fractional $p$-Laplacian equation in the Fast Diffusion Range

We establish existence, uniqueness as well as quantitative estimates for solutions to the fractional nonlinear diffusion equation, $\partial_t u +{\mathcal L}_{s,p} (u)=0$, where ${\mathcal L}_{s,p}=(-\Delta)_p^s$ 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 $sp<1$. The equation is posed in the whole space $x\in {\mathbb R}^N$. 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 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 analyze the conditions for extinction in finite time.