The influence of viscous dissipations on the nonlinear acoustic wave equation with second sound

We study the effect of a viscous dissipation on the Cauchy problem for a Cattaneo-type model in nonlinear acoustics, established by applying the Lighthill approximation for the viscous or inviscid fluid model. The contribution of this paper is twofold. For the nonlinear viscous Cattaneo-type model involving a fractional Laplacian $(-\Delta)^{\alpha}$ in the viscous damping with $\alpha\in[0,1]$, we derive optimal decay rates for global (in time) solutions with small data in certain Sobolev spaces. Furthermore, by introducing a threshold $\alpha=1/2$ for the power of the fractional viscous dissipation, we derive an anomalous diffusion profile when $\alpha\in[0,1/2)$ and a diffusion wave profile when $\alpha\in[1/2,1]$ for large-time. Whereas, for the nonlinear inviscid Cattaneo-type model (or the Jordan-Moore-Gibson-Thompson equation in the critical case), we obtain the blow-up of the energy solutions in finite time under suitable assumptions for the initial data. Thus, the presence of a viscous dissipation in the nonlinear Cattaneo-type model is a criterion for the global (in time) existence and blow-up of solutions.
Comment: Blow-up for the inviscid model (or the JMGT equation in the critical case) is added