@Article{CMAA-1-3,
author = {Li, Dong and Quan, Chaoyu and Tao, Tang and Yang, Wen},
title = {Sharp Convergence to Steady States of Allen-Cahn},
journal = {Communications in Mathematical Analysis and Applications},
year = {2022},
volume = {1},
number = {3},
pages = {355--394},
abstract = {<p style="text-align: justify;">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&nbsp;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.</p>},
issn = {2790-1939},
doi = {https://doi.org/2022-CMAA-20661},
url = {https://global-sci.com/article/81434/sharp-convergence-to-steady-states-of-allen-cahn}
}