*Accepted Paper*

**Inserted:** 18 nov 2005

**Last Updated:** 13 dec 2006

**Journal:** Discrete Cont. Dyn. Syst.

**Volume:** 16

**Number:** 4

**Pages:** 783-842

**Year:** 2006

**Abstract:**

We study the gradient dynamics associated with the functional $F_\phi(u) := \frac{1}{2}\int_{I} \phi(u_x) dx$ and with its singular perturbation $F_\phi^\epsilon(u):=\frac{1}{2}\int_I \left(\epsilon^2 ({u_{xx}})^2 + \phi(u_x)\right) dx$, where $\phi$ is {\it non convex}. We discuss, with the support of numerical simulations, various aspects of the global dynamics of solutions $u^\epsilon$ of the singularly perturbed equation $u_t = - \epsilon^2 u_{xxxx} + \frac{1}{2} \phi''(u_x)u_{xx}$ for small values of $\epsilon >0$. We develop a point of view that leads to a reinterpretation of the unperturbed equation $u_t = \frac{1}{2} (\phi'(u_x))_x$, and to a well defined notion of solution which is conjectured to coincide with the limit of $u^\epsilon$ as $\epsilon \to 0^+$.