A model problem for unsaturated porous media flow with dynamic capillary pressure

We consider a simplified model of vertical non-static groundwater flow, resulting in a Burgers' diffusive equation extended with a third order term with mixed derivatives in space and time. This model is motivated by previous work on the existence of travelling wave solutions on a more general model. We investigate stability of travelling wave solutions of the simplified model, for which we first obtain well-posedness results.