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 2009 |
| Externally published | Yes |
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Keywords
- Allen-Cahn equation
- Almost periodic
- Asymptotic stability
- Planar wave
ASJC Scopus subject areas
- Analysis
- Applied Mathematics
Cite this
Stability of planar waves in the Allen-Cahn equation. / Matano, Hiroshi; Nara, Mitsunori; Taniguchi, Masaharu.
In: Communications in Partial Differential Equations, Vol. 34, No. 9, 09.2009, p. 976-1002.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Stability of planar waves in the Allen-Cahn equation
AU - Matano, Hiroshi
AU - Nara, Mitsunori
AU - Taniguchi, Masaharu
PY - 2009/9
Y1 - 2009/9
N2 - 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.
AB - 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.
KW - Allen-Cahn equation
KW - Almost periodic
KW - Asymptotic stability
KW - Planar wave
UR - http://www.scopus.com/inward/record.url?scp=70449470146&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=70449470146&partnerID=8YFLogxK
U2 - 10.1080/03605300902963500
DO - 10.1080/03605300902963500
M3 - Article
AN - SCOPUS:70449470146
VL - 34
SP - 976
EP - 1002
JO - Communications in Partial Differential Equations
JF - Communications in Partial Differential Equations
SN - 0360-5302
IS - 9
ER -