Growing solutions of the fractional [formula omitted]-Laplacian equation in the Fast Diffusion range

Abstract

We establish existence, uniqueness as well as quantitative estimates for solutions u(t,x) to the fractional nonlinear diffusion equation, ∂tu+Ls,p(u)=0, where Ls,p=(−Δ)ps 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∈RN. 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.