Existence of Undercompressive Traveling Waves in Thin Film Equations

We consider undercompressive traveling wave solutions of the partial differential equation ∂th+ ∂xf(h) = −∂x(h∂ xh) +D∂x(h∂xh), when the flux function f has the nonconvex form f(h) = h2 − h3. In numerical simulations, these waves appear to play a central role in the dynamics of the PDE; they also explain unusual phenomena in experiments of driven contact lines modeled by the PDE. We prove existence of an undercompressive traveling wave solution for sufficiently small nonnegative D and nonexistence when D is sufficiently large.