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In this paper, we consider several time-fractional generalizations of the Jordan–Moore–Gibson–Thompson (JMGT) equations in nonlinear acoustics as well as their linear Moore–Gibson–Thompson (MGT) versions. Following the procedure described in Jordan (2014), these time-fractional acoustic equations are derived from four fractional versions of the Maxwell–Cattaneo law in Compte and Metzler (1997). Additionally to providing well-posedness results for each of them, we also study the respective limits as the fractional order tends to one, leading to the classical third order in time (J)MGT equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Jordan–Moore–Gibson–Thompson equation, a third-order in time wave equation describing the nonlinear propagation of sound that avoids the infinite signal speed paradox of classical second-order in time strongly damped models of nonlinear acoustics, such as the Westervelt and the Kuznetsov equation. We show well-posedness in an acoustic velocity potential formulation with and without gradient nonlinearity, corresponding to the Kuznetsov and the Westervelt nonlinearities, respectively. Moreover, we consider the limit as the parameter of the third-order time derivative that plays the role of a relaxation time tends to zero, which again leads to the classical Kuznetsov and Westervelt models. To this end, we establish appropriate energy estimates for the linearized equations and employ fixed-point arguments for well-posedness of the nonlinear equations. The theoretical results are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

This work is concerned with the study of fundamental models from nonlinear acoustics. In Part I, a hierarchy of nonlinear damped wave equations arising in the description of sound propagation in thermoviscous fluids is deduced. In particular, a rigorous justification of two classical models, the Kuznetsov and Westervelt equations, retained as limiting systems for vanishing thermal conductivity and consistent initial data, is given. Numerical comparisons that confirm and complement the theoretical results are provided in Part II. [ABSTRACT FROM AUTHOR]

We consider a third order in time equation which arises, e.g. as a model for wave propagation in viscous thermally relaxing fluids. This equation displays, even in the linear version, a variety of dynamical behaviors for its solution that depend on the physical parameters in the equation. These range from non-existence and instability to exponential stability (in time) as was shown for the constant coefficient case in Ref. 23. In case of vanishing diffusivity of the sound, there is a lack of generation of a semigroup associated with the linear dynamics. If diffusivity of the sound is positive, the linear dynamics is described by a strongly continuous hyperbolic-like evolution. This evolution is exponentially stable provided sufficiently large viscous damping is accounted for in the model. In this paper, we consider the full nonlinear model referred to as Jordan-Moore-Gibson-Thompson equation. This model can be seen as a "hyperbolic" version of Kuznetsov's equation, where the linearization of the latter corresponds to an analytic semigroup. This is no longer valid for the presently considered third-order model whose linearization is associated with a group structure. In order to carry out the analysis of the nonlinear model, we first consider time and space-dependent viscosity which then leads to evolution rather than semigroup generators. Decay rates for both "natural" and "higher" level energies are derived. Relevant physical parameters that are responsible for spectral behavior (continuous and point spectrum) are identified. The theoretical estimates proved in the paper are confirmed by numerical simulations. The derived energy estimates are then used in order to establish global well-posedness and exponential decay for the solutions to the nonlinear equation. [ABSTRACT FROM AUTHOR]