The authors study the gradient flow associated with the functional \(F_{\phi} (u): = \frac12 \int_I \phi (u_x)\, dx\), where \(\phi\) is non convex, and with its singular perturbation \(F_{\phi}^x (u): = \frac12 \int_I (\varepsilon^2 (u_{xx})^2 + \phi (u_x))\,dx\). With the support of numerical simulations, various aspects of the global dynamics of solutions \(u^{\varepsilon}\) of the singularly perturbed equation \(u_t= -\varepsilon^2 u_{xxxx} +\frac12 \phi'' (u_x) u_{xx}\) for small values of \(\varepsilon > 0\) are discussed. Their analysis leads to a reinterpretation of the unperturbed equation \(u_{tt} =\frac12 (\phi' (u_x))_x\), and to a well defined notion of a solution. Examine the conjecture that this solution coincides with the limit of \(u^{\varepsilon}\) as \(\varepsilon \to 0^+\) is given.