Stability of non-isolated asymptotic profiles for fast diffusion

Abstract

The stability of asymptotic profiles of solutions to the Cauchy-Dirichlet problem for Fast Diffusion Equation (FDE, for short) is discussed. The main result of the present paper is the stability of any asymptotic profiles of least energy. It is noteworthy that this result can cover non-isolated profiles, e.g., those for thin annular domain cases. The method of proof is based on the Lojasiewicz-Simon inequality, which is usually used to prove the convergence of solutions to prescribed limits, as well as a uniform extinction estimate for solutions to FDE. Besides, local minimizers of an energy functional associated with this issue are characterized. Furthermore, the instability of positive radial asymptotic profiles in thin annular domains is also proved by applying the Lojasiewicz-Simon inequality in a different way.

References in corpus (1)

• Asymptotics of the fast diffusion equation via entropy estimates

Cited by in corpus (4)

• An Energetic Variational Approach for the Cahn--Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis
• Asymptotics near extinction for nonlinear fast diffusion on a bounded domain
• Rates of convergence to non-degenerate asymptotic profiles for fast diffusion via energy methods
• The Cauchy-Dirichlet Problem for Singular Nonlocal Diffusions on Bounded Domains