Continua of local minimizers in a quasilinear model of phase transitions

Pavel Drábek; Stephen Robinson

doi:10.3934/dcds.2013.33.163

Article Contents

2013, Volume 33, Issue 1: 163-172. Doi: 10.3934/dcds.2013.33.163

This issue Previous Article Some bifurcation results for rapidly growing nonlinearities Next Article A class of singular first order differential equations with applications in reaction-diffusion

Continua of local minimizers in a quasilinear model of phase transitions

Pavel Drábek^1, and
Stephen Robinson^2,

1.
Department of Mathematics and Center N.T.I.S., University of West Bohemia, P.O. Box 314,306 14 Pilsen
2.
Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109

Received: July 2011

Revised: February 2012

Published: January 2013

Abstract / Introduction Related Papers Cited by

Abstract

In this paper we study critical points of the functional \begin{eqnarray*} J_{\epsilon}(u):= \frac{\epsilon^p}{p}\int_0^1|u_x|^pdx+\int_0^1F(u)dx, \; u∈w^{1,p}(0,1), \end{eqnarray*} where F:$\mathbb{R}$→$\mathbb{R}$ is assumed to be a double-well potential. This functional represents the total free energy in phase transition models. We consider a non-classical choice for $F$ modeled on $F(u)=|1-u^2|^{\alpha}$ where $1< \alpha < p$. This choice leads to the existence of multiple continua of critical points that are not present in the classical case $\alpha= p = 2$. We prove that the interior of these continua are local minimizers. The energy of these local minimizers is strictly greater than the global minimum of $J_{\epsilon}$. In particular, the existence of these continua suggests an alternative explanation for the slow dynamics observed in phase transition models.

Keywords:

Mathematics Subject Classification: Primary: 34B15, 35K55; Secondary: 34B16, 49R05.

Citation:

Related Papers

Cited by

References

[1]	J. Carr and R. L. Pego, Metastable patterns in solutions of u _t = ε² $u_{x x}$ - f(u), Comm. Pure Appl. Math., 42 (1989), 523-576.doi: 10.1002/cpa.3160420502.
[2]	P. Drábek, A. Kufner and F. Nicolosi, "Quasilinear Elliptic Equations with Degenerations and Singularities," De Gruyter Series in Nonlinear Analysis and Applications 5, Walter de Gruyter, Berlin, New York, 1997.
[3]	P. Drábek, R. Manásevich and P. Takáč, Slow dynamics in a quasilinear model for phase transitions in one space dimension, In "Nonlinear Elliptic Partial Differential Equations," Workshop in celebration of Jean-Pierre Gossez's 65th birthday September 24, 2009, Brusels, Contemporary Mathematics Series, 540, 95-134.
[4]	P. Drábek and S. Robinson, Continua of local minimizers in a non-smooth model of phase transitions, Z. Angew. Math. Phys., 62 (2011), 609-622.