The Kuramoto-Velarde equation describes the spatio-temporal evolution of step morphology on crystal surfaces, as well as the dynamics of spinodal decomposition in phase-separating systems subjected to an external field. In this paper, we prove the well-posedness of the solutions for the initial-boundary value problem for this equation, under several possible boundary conditions.

This paper modeled the dynamics of microbubbles coated with viscoelastic shells using the modified Korteweg-de Vries-Burgers equation, a nonlinear third-order partial differential equation. This study focused on the well-posedness of the Cauchy problem associated with this equation.