Abstract
We study the asymptotic stability of planar waves for the Allen-Cahn equation on ℝn, where n ≥ 2. Our first result states that planar waves are asymptotically stable under any-possibly large-initial perturbations that decay at space infinity. Our second result states that the planar waves are asymptotically stable under almost periodic perturbations. More precisely, the perturbed solution converges to a planar wave as t → ∞. The convergence is uniform in ℝn. Lastly, the existence of a solution that oscillates permanently between two planar waves is shown, which implies that planar waves are not asymptotically stable under more general perturbations.
| Original language | English |
|---|---|
| Pages (from-to) | 976-1002 |
| Number of pages | 27 |
| Journal | Communications in Partial Differential Equations |
| Volume | 34 |
| Issue number | 9 |
| DOIs | |
| Publication status | Published - Sep 1 2009 |
| Externally published | Yes |
Keywords
- Allen-Cahn equation
- Almost periodic
- Asymptotic stability
- Planar wave
ASJC Scopus subject areas
- Analysis
- Applied Mathematics
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Cite this
Matano, H., Nara, M., & Taniguchi, M. (2009). Stability of planar waves in the Allen-Cahn equation. Communications in Partial Differential Equations, 34(9), 976-1002. https://doi.org/10.1080/03605300902963500