  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.
