Date of Award
Doctor of Philosophy
The mathematical topic of the present thesis is a third-order (in time) semilinear Partial Differential Equation (PDE), generally referred to, in the context of acoustics, as the Jordan-Moore-Gibson-Thompson equation. It arises in a variety of physical contexts such as in describing the behavior of viscoelastic materials and propagation of acoustic waves. The presence of the third-order time derivative is due to the use of the second-sound phenomenon as a model for heat propagation. This anticipates thermal waves traveling with finite velocity, to replace the classical thermal theory, which is based on diffusion and under which heat signals have an infinite speed of propagation. In the first part of the thesis we study the singular effects of the thermal relaxation time parameter in the propagation of nonlinear acoustic waves. We find that the comparable quantities of the nonlinear parabolic dynamics (with zero relaxation) can be viewed as strong limits of the hyperbolic dynamics in environments where thermal relaxation is small. The inherent singularity does not allow classical convergence theories to apply even in the linear case. In the nonlinear case, the assumption of smallness of the initial data prevents arguments on density of smooth solutions to be successful. Tight control of the smallness of the initial data, which is then propagated through the dynamics, allows establishing strong convergence of the respective semigroups. This result solves an open problem raised recently in the literature. In the second part of the thesis we study the effects of boundary dissipation, imposed on a suitable portion of the boundary of a bounded domain with smooth boundary, in the asymptotic (in time) behavior of global nonlinear acoustic waves solutions with emphasis on the critical case; that is, when all natural frictional (viscoelastic) interior damping is assumed to vanish. We show that boundary feedback dissipation yields exponential stability. The difficulty here is the avoidance of degeneracies due to nonlinearities. Uniform stability of the corresponding linearization is pivotal and enables to construct fixed-point global-in-time solutions for each boundary configuration. Hence, stability results, uniform with respect to viscoelastic effects, are derived to accommodate not only the critical case but also cases where interior measurements can only be performed in subsets (perhaps even discrete points) of the domain.
Dissertation or thesis originally submitted to ProQuest
dos Santos Bongarti, Marcelo Adriano, "ON NONLINEAR ACOUSTIC WAVES OF THIRD--ORDER (IN TIME) AND APPLICATIONS" (2021). Electronic Theses and Dissertations. 2518.