Nonlinear acoustics: Blackstock-Crighton equations with a periodic forcing term

The 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 arbitrary 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 three-dimensional 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.