Stability of under-compressive waves with second and fourth order diffusions

This is a continuation of our work on the nonlinear stability of traveling shock fronts arising in multidimensional conservation laws with fourth order regularization only. Our motivating example is the thin film equation for which planar waves correspond with fluid coating a pre-wetted surface. Under only the fourth order regularization, we established the nonlinear stability of compressive waves for dimensions $d\geq 2$, and of under-compressive waves for dimensions $d\geq 3$ under general spectral conditions. The case of stability for under-compressive waves in the thin film equations for the critical dimensions $d=1,2$ remained open. In this paper we study the nonlinear stability of under-compressive waves by assuming both the second and fourth order regularization. We present a step toward the open problem by establishing the nonlinear stability of under-compressive waves in dimensions $d\geq 2$ under general spectral conditions. We emphasize the above mentioned stability question remains still open.