Nonlinear Acoustics: Blackstock–Crighton Equations with a Periodic Forcing Term

Blackstock–Crighton equations describe the motion of a viscous, heat-conducting, compressible fluid. They are used as models for acoustic wave propagation in a medium in which both nonlinear and dissipative effects are taken into account. In this article, a mathematical analysis of the Blackstock–Crighton equations with a time-periodic forcing term is carried out. For time-periodic data sufficiently restricted in size it is shown that a time-periodic solution of the same period always exists. This implies that the dissipative effects are sufficient to avoid resonance within the Blackstock–Crighton models. The equations are considered in a bounded domain with both non-homogeneous Dirichlet and Neumann boundary values. Existence of a solution is obtained via a fixed-point argument based on appropriate a priori estimates for the linearized equations.