Year: 2022
Author: Dong Li, Chaoyu Quan, Tao Tang, Wen Yang
Communications in Mathematical Analysis and Applications, Vol. 1 (2022), Iss. 3 : pp. 355–394
Abstract
In our recent work we found a surprising breakdown of symmetry conservation: using standard numerical discretization with very high precision the computed numerical solutions corresponding to very nice initial data may converge to completely incorrect steady states due to the gradual accumulation of machine round-off error. We solved this issue by introducing a new Fourier filter technique for solutions with certain band gap properties. To further investigate the attracting basin of steady states we classify in this work all possible bounded nontrivial steady states for the Allen-Cahn equation. We characterize sharp dependence of nontrivial steady states on the diffusion coefficient and prove strict monotonicity of the associated energy. In particular, we establish a certain self-replicating property amongst the hierarchy of steady states and give a full classification of their energies and profiles. We develop a new modulation theory and prove sharp convergence to the steady state with explicit rates and profiles.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2022-CMAA-20661
Communications in Mathematical Analysis and Applications, Vol. 1 (2022), Iss. 3 : pp. 355–394
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 40
Keywords: Allen-Cahn equation steady state ground state solution asymptotic behavior.