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

We study the gradient flow associated with the functional F-phi(u) := 1/2 integral(I) phi(u(x)) dx, where phi is non convex, and with its singular perturbation F-phi(epsilon)(u) := 1/2 integral(I) (epsilon(2)(u(xx))(2) + phi(u(x)))dx. 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) + 1/2 phi''(u(x)) u(xx) for small values of epsilon > 0. Our analysis leads to a reinterpretation of the unperturbed equation u(t) = 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(epsilon) as epsilon --> 0(+).