×

zbMATH — the first resource for mathematics

Anomalous scaling for three-dimensional Cahn-Hilliard fronts. (English) Zbl 1078.35049
Summary: We prove the stability of the one-dimensional kink solution of the Cahn-Hilliard equation under \(d\)-dimensional perturbations for \(d \geqslant 3\). We also establish a novel scaling behavior of the large-time asymptotics of the solution. The leading asymptotics of the solution is characterized by a length scale proportional to \(t^{1/3}\) instead of the usual \(t^{1/2}\) scaling typical to parabolic problems.

MSC:
35K55 Nonlinear parabolic equations
35K25 Higher-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bettinson, Phys Rev E 54 pp 6102– (1996)
[2] Bricmont, Comm Pure Appl Math 47 pp 893– (1994)
[3] Bricmont, Comm Pure Appl Math 52 pp 839– (1999)
[4] Carlen, Comm Math Phys 224 pp 323– (2001)
[5] ; Instabilities and Fronts in Extended Systems. Princeton University, Princeton, N.J., 1990. · Zbl 0732.35074
[6] Stability of Cahn-Hilliard fronts in three dimensions. Doctoral dissertation, University of Helsinki, 2003.
[7] ; Quantum Mechanics. 3rd ed. Pergamon, New York, 1981.
[8] Pego, Phil Trans R Soc Lond A 340 pp 47– (1992)
[9] Shinozaki, Phys Rev E 47 pp 804– (1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.