Thin films of silicon oil driven up an inclined silicon wafer by a thermally induced Marangoni force develop unusual shock profiles involving a non-classical undercompressive shock, if the counteracting parallel component of gravity is sufficiently large (Bertozzi et al 1998 Phys. Rev. Lett. 81 5169-72). They arise as a result of the interaction of a non-convex flux with the fourth-order diffusion generated by surface tension. In this work, we investigate how the dynamical behaviour of the solution is affected by including second-order diffusion resulting from the normal component of gravity; this component was neglected in the previous study. Then the governing equation for the film profile h (x , t ) becomes h_t + ( h ² - h ³ )_x = - ( h ³ h_xxx )_x + D ( h ³ h_x )_x               D 0. The numerical simulations in this paper confirm that neglecting second-order diffusion is justified for small D , but find that for larger D , the structure of the solution changes dramatically. We give a detailed account of the transitions that occur while increasing D and make predictions for future experiments carried out at small inclination angles, corresponding to moderately large D .