Asymptotic behaviors for Blackstock's model of thermoviscous flow

We study a fundamental model in nonlinear acoustics, precisely, the general Blackstock model (i.e., without Becker’s assumption) in the whole space Rn. This model describes nonlinear acoustics in perfect gases under the irrotational flow. By means of the Fourier analysis we will derive L2 estimates for the solution of the linear homogeneous problem and its derivatives. Then, we will apply these estimates to study three different topics: the optimality of the decay estimates in the case n≥5 and the optimal growth rate for the L2-norm of the solution for n = 3, 4; the singular limit problem in determining the first- and second-order profiles for the solution of the linear Blackstock model with respect to the small thermal diffusivity; the proof of the existence of global (in time) small data Sobolev solutions with suitable regularity for a nonlinear Blackstock model.