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1 December 2002 Stable patterns for fourth-order parabolic equations
R. C. Vandervorst, J. B. van den Berg
Duke Math. J. 115(3): 513-558 (1 December 2002). DOI: 10.1215/S0012-7094-02-11534-8

Abstract

We consider fourth-order parabolic equations of gradient type. For the sake of simplicity, the analysis is carried out for the specific equation $u\sb t=-\gamma\ u\sb {xxxx}+\beta u\sb {xx}-F\sp \prime(u)$ with $(t,x)\in (0,\infty)\times(0, L)$ and $\gamma,\beta>0$ and where $F(u)$ is a bistable potential. We study its stable equilibria as a function of the ratio $\gamma/beta\sp 2$. As the ratio $\gamma/beta\sp 2$ crosses an explicit threshold value, the number of stable patterns grows to infinity as $L\to \infty$. The construction of the stable patterns is based on a variational gluing method that does not require any genericity conditions to be satisfied.

Citation

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R. C. Vandervorst. J. B. van den Berg. "Stable patterns for fourth-order parabolic equations." Duke Math. J. 115 (3) 513 - 558, 1 December 2002. https://doi.org/10.1215/S0012-7094-02-11534-8

Information

Published: 1 December 2002
First available in Project Euclid: 26 May 2004

zbMATH: 1020.35027
MathSciNet: MR1940411
Digital Object Identifier: 10.1215/S0012-7094-02-11534-8

Subjects:
Primary: 35K35
Secondary: 35A15 , 35K55 , 37G35 , 37J45

Rights: Copyright © 2002 Duke University Press

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Vol.115 • No. 3 • 1 December 2002
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