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

We study the gradient flow associated with the functional Fϕ(u) := 12∫Iϕ(ux) dx, where ϕ is non convex, and with its singular perturbation Fεϕ(u):=12∫I(ε2(uxx)2+ϕ(ux))dx. We discuss, with the support of numerical simulations, various aspects of the global dynamics of solutions uε of the singularly perturbed equation ut=−ε2uxxxx+12ϕ′′(ux)uxx for small values of ε>0. Our analysis leads to a reinterpretation of the unperturbed equation ut=12(ϕ′(ux))x, and to a well defined notion of a solution. We also examine the conjecture that this solution coincides with the limit of uε as ε→0+.