Optimal decay rates and asymptotic profiles for the nonlinear acoustic wave equation with fractional Laplacians

In this paper, we study the Cauchy problem for the nonlinear acoustic wave equation with the Cattaneo law involving fractional Laplacians $(-\Delta)^{\alpha}$ of the viscosity with $\alpha\in[0,1]$, which is established by applying the Lighthill approximation of the fractional Navier-Stokes-Cattaneo equations under irrotational flows. Exploring structures of the nonlinearities, we rigorously demonstrate optimal decay rates of the global (in time) small datum Sobolev solutions with suitable regularities. Furthermore, by introducing a threshold $\alpha=1/2$, we derive the anomalous diffusion profiles when $\alpha\in[0,1/2)$ and the diffusion wave profiles when $\alpha\in[1/2,1]$ as large time. These results show influences of the fractional index $\alpha$ on large time behaviors of the solutions.