Stability of Non-Isolated Asymptotic Profiles for Fast Diffusion

Mathematics - Analysis of PDEs 35K67, 35B40, 35B35 FOS: Mathematics FOS: Physical sciences Mathematical Physics (math-ph) 0101 mathematics 16. Peace & justice 01 natural sciences Mathematical Physics Analysis of PDEs (math.AP)
DOI: 10.1007/s00220-016-2649-0 Publication Date: 2016-05-27T10:23:42Z
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.
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