Large-time asymptotic behaviors for linear Blackstock's model of thermoviscous flow

Abstract

In the classical theory of acoustic waves, Blackstock's model was proposed in 1963 to characterize the propagation of sound in thermoviscous fluids. In this paper, we investigate large-time asymptotic behaviors of the linear Cauchy problem for general Blackstock's model (that is, without Becker's assumption on monatomic perfect gases). We derive first- and second-order asymptotic profiles of solution as $t\gg1$ by applying refined WKB analysis and Fourier analysis. Our results not only improve optimal estimates in [Chen-Ikehata-Palmieri, \emph{Indiana Univ. Math. J.} (2023)] for lower dimensional cases, but also illustrate the optimal leading term and novel second-order profiles of solution with additional weighted $L^1$ data.