Stability analysis of asymptotic profiles for sign-changing solutions to fast diffusion equations

Abstract

Every solution u = u(x, t) of the Cauchy–Dirichlet problem for the fast diffusion equation, ∂ t (|u| m-2 u) = Δu in Ω × (0, ∞) with a smooth bounded domain Ω of $${\mathbb{R}^N}$$ and 2 < m < 2* : = 2N/(N − 2)+, vanishes in finite time at a power rate. This paper is concerned with asymptotic profiles of sign-changing solutions and a stability analysis of the profiles. Our method of proof relies on a detailed analysis of a dynamical system on some surface in the usual energy space as well as energy method and variational method.