On some methods in nonlinear partial differential equations and their applications to the Blackstock-Crighton-Westervelt rotational model equation

We consider the Blackstock--Crighton rotational model equation which is a fourth-order in space and third-order in time nonlinear partial differential equation and arises in the context of the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids. We use the theory of operator semigroups to investigate the linearization of the equation and show that the underlying semigroup is analytic which, together with a negative spectral bound of its generator, leads to exponential decay results for the linear homogeneous equation. Moreover, invoking the Banach fixed-point theorem, we prove local well-posedness of the Blackstock--Crighton model for sufficiently small initial data. Finally, we show how barrier's method is used to obtain global in time well-posedness and provide exponential decay results also for the nonlinear equation.