Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping

Presented here is a study of a viscoelastic wave equation with supercritical source and damping terms. We employ the theory of monotone operators and nonlinear semigroups, combined with energy methods to establish the existence of a unique local weak solution. In addition, it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense.Comment: The 2nd version includes a new proof of the energy identit

Finite dimensional attractor for a composite system of wave/plate equations with localised damping

The long-term behaviour of solutions to a model for acoustic-structure interactions is addressed; the system is comprised of coupled semilinear wave (3D) and plate equations with nonlinear damping and critical sources. The questions of interest are: existence of a global attractor for the dynamics generated by this composite system, as well as dimensionality and regularity of the attractor. A distinct and challenging feature of the problem is the geometrically restricted dissipation on the wave component of the system. It is shown that the existence of a global attractor of finite fractal dimension -- established in a previous work by Bucci, Chueshov and Lasiecka (Comm. Pure Appl. Anal., 2007) only in the presence of full interior acoustic damping -- holds even in the case of localised dissipation. This nontrivial generalization is inspired by and consistent with the recent advances in the study of wave equations with nonlinear localised damping.Comment: 40 pages, 1 figure; v2: added references for Section 1, submitte

Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions

The goal of this work is to study a model of the wave equation with dynamic boundary conditions and a viscoelastic term. First, applying the Faedo-Galerkin method combined with the fixed point theorem, we show the existence and uniqueness of a local in time solution. Second, we show that under some restrictions on the initial data, the solution continues to exist globally in time. On the other hand, if the interior source dominates the boundary damping, then the solution is unbounded and grows as an exponential function. In addition, in the absence of the strong damping, then the solution ceases to exist and blows up in finite time.Comment: arXiv admin note: text overlap with arXiv:0810.101

ON NONLINEAR ACOUSTIC WAVES OF THIRD--ORDER (IN TIME) AND APPLICATIONS

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

Regular global attractors for wave equations with degenerate memory

We consider the wave equation with degenerate viscoelastic dissipation recently examined in Cavalcanti, Fatori, and Ma, Attractors for wave equations with degenerate memory, J. Differential Equations (2016). Under some additional assumptions, we show the existence of a compact absorbing set. This result provides further regularity for the global attractor and shows that it consists of regular solutions