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Boundary feedback stabilization of a critical third–order (in time) semilinear Jordan–Moore–Gibson–Thompson (JMGT) is considered. The word critical here refers to the usual case where media–damping effects are non–existent or non–measurable and therefore cannot be relied upon for stabilization purposes. Motivated by modeling aspects in high-intensity focused ultrasound (HIFU) technology, the boundary feedback under consideration is supported only on a portion of the boundary. At the same time, the remaining part is undissipated and subject to Neumann/Robin boundary conditions. As such, unlike Dirichlet, it fails to satisfy the Lopatinski condition, a fact which compromises tangential regularity on the boundary [37]. In such a configuration, the analysis of uniform stabilization from the boundary becomes subtle and requires careful geometric considerations and microlocal analysis estimates. The nonlinear effects in the model demand construction of suitably small solutions which are invariant under the dynamics. The assumed smallness of the initial data is required only at the lowest energy level topology, which is sufficient to construct sufficiently smooth solutions to the nonlinear model. 

In this paper, we use modified versions of the SIAR model for epidemics to propose two ways of understanding and quantifying the effect of non-compliance to non-pharmaceutical intervention measures on the spread of an infectious disease. The SIAR model distinguishes between symptomatic infected (I) and asymptomatic infected (A) populations. One modification, which is simpler, assumes a known proportion of the population does not comply with government mandates such as quarantining and social-distancing. In a more sophisticated approach, the modified model treats non-compliant behavior as a social contagion. We theoretically explore different scenarios such as the occurrence of multiple waves of infections. Local and asymptotic analyses for both models are also provided. 

The Jordan–Moore–Gibson–Thompson (JMGT) equation is a well-established and recently widely studied model for nonlinear acoustics (NLA). It is a third–order (in time) semilinear Partial Differential Equation (PDE) with a distinctive feature of predicting the propagation of ultrasound waves at finite speed. This is due to the heat phenomenon known as second sound which leads to hyperbolic heat-wave propagation. In this paper, we consider the problem in the so called "critical" case, where free dynamics is unstable. In order to stabilize, we shall use boundary feedback controls supported on a portion of the boundary only. Since the remaining part of the boundary is not "controlled", and the imposed boundary conditions of Neumann type fail to saitsfy Lopatinski condition, several mathematical issues typical for mixed problems within the context o boundary stabilizability arise. To resolve these, special geometric constructs along with sharp trace estimates will be developed. The imposed geometric conditions are motivated by the geometry that is suitable for modeling the problem of controlling (from the boundary) the acoustic pressure involved in medical treatments such as lithotripsy, thermotherapy, sonochemistry, or any other procedure involving High Intensity Focused Ultrasound (HIFU).