  In this paper we focus on a small amplitude approximation of a
Navier-Stokes-Fourier system modeling nonlinear acoustics. Omitting all third
and higher order terms with respect to certain small parameters, we obtain a
first order in time system containing linear and quadratic pressure and
velocity terms.
  Subsequently, the well-posedness of the derived system is shown using the
classical method of Galerkin approximation in combination with a fixed point
argument. We first prove the well-posedness of a linearized equation using
energy estimates and then the well-posedness of the nonlinear system using a
Newton-Kantorovich type argument. Based on this, we also obtain global in time
well-posedness for small enough data and exponential decay. This is in line
with semigroup results for a linear part of the system that we provide as well.
