Existence of undercompressive traveling waves in thin film equations

We consider undercompressive traveling wave solutions of the partial differential equation partial derivative(t)h + partial derivative(x)f (h) = -partial derivative(x) (h(3)partial derivative(x)(3)h) + D partial derivative(x) (h(3)partial derivative(x)h), when the flux function f has the nonconvex form f (h) = h(2) - h(3). 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.