, Volume 141, Issue 3-4, pp 559-587

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

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Abstract

Every solution uu(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.