A concept of solution and numerical experiments for forward-backward diffusion equations

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