## Advances in Differential Equations

- Adv. Differential Equations
- Volume 2, Number 1 (1997), 1-38.

### Singular limits of scalar Ginzburg-Landau equations with multiple-well potentials

#### Abstract

We characterize the limiting behavior of scalar phase-field equations with infinitely many potential wells as the density of potential wells tends to infinity. An example of such a family of equations is \[ \mathcal{u}^e_t = \Delta\mathcal{u}^e - \frac1{\epsilon^{1+\alpha}}W'(\frac{\mathcal{u}^e}{\epsilon^{1-\alpha}}), \] where $W$ is a periodic function. We prove that solutions of the above equation converge to solutions of the Mean Curvature partial differential equation for a range of positive values of the parameter $\alpha$, and we also determine the limiting equation when $\alpha = 0$. We show that our techniques can be modified to apply to fully nonlinear equations and to other classes of infinite-well equations. We discuss some applications to questions of interaction between wave fronts in dynamic phase transitions.

#### Article information

**Source**

Adv. Differential Equations Volume 2, Number 1 (1997), 1-38.

**Dates**

First available in Project Euclid: 24 April 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1366809227

**Mathematical Reviews number (MathSciNet)**

MR1424762

**Zentralblatt MATH identifier**

1023.35526

**Subjects**

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Secondary: 35B40: Asymptotic behavior of solutions

#### Citation

Jerrard, Robert L. Singular limits of scalar Ginzburg-Landau equations with multiple-well potentials. Adv. Differential Equations 2 (1997), no. 1, 1--38. https://projecteuclid.org/euclid.ade/1366809227