A Class of Sixth Order Viscous Cahn-Hilliard Equation with Willmore Regularization in Double-struck capital R3

The main purpose of this paper is to study the Cauchy problem of sixth order viscous Cahn-Hilliard equation with Willmore regularization. Because of the existence of the nonlinear Willmore regularization and complex structures, it is difficult to obtain the suitable a priori estimates in order to prove the well-posedness results, and the large time behavior of solutions cannot be shown using the usual Fourier splitting method. In order to overcome the above two difficulties, we borrow a fourth-order linear term and a second-order linear term from the related term, rewrite the equation in a new form, and introduce the negative Sobolev norm estimates. Subsequently, we investigate the local well-posedness, global well-posedness, and decay rate of strong solutions for the Cauchy problem of such an equation in R3, respectively.