Uniqueness and Explicit Solutions of a Mixed-Type Equation in the de Gennes Model

Abstract

This paper considers an initial boundary value problem for the linear wave equation in smectics in the de Gennes model, i.e., for a linear partial differential equation of mixed type with constant coefficients. For the original equation, we derive an integral identity that implies the energy conservation law, which is used to prove the uniqueness of a solution under homogeneous Dirichlet boundary conditions and zero initial conditions. For a rectangular parallelepiped domain with Dirichlet boundary conditions, an explicit solution is constructed using separation of variables. The solution is represented as a multidimensional Fourier series in terms of sine functions. The dispersion relation and formulas for the Fourier coefficients are written out in terms of initial data. The dimension \(n = 4\) is considered for the radially symmetric case in a cylindrical domain (with an annulus base). In this case, using the substitution \(u = {v}{\text{/}}r\) simplifies the equation, the singularity at zero disappears, and the radial part of the solution is described by ordinary sines, rather than by Bessel functions. Explicit formulas for the eigenfrequencies and eigenfunctions are derived, and a series representation of the solution is obtained.