Growing solutions of the fractional p-Laplacian equation in the Fast Diffusion range. (January 2022) - Vázquez, Juan Luis
Abstract: We establish existence, uniqueness as well as quantitative estimates for solutions u ( t, x ) to the fractional nonlinear diffusion equation, ∂ t u + L s, p ( u ) = 0, where L s, p = ( − Δ ) 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 s p < 1 . The equation is posed in the whole space x ∈ 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.