Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions

In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the Kelvin-Voigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained

Optimal rates of decay for operator semigroups on Hilbert spaces

We investigate rates of decay for $C_0$-semigroups on Hilbert spaces under assumptions on the resolvent growth of the semigroup generator. Our main results show that one obtains the best possible estimate on the rate of decay, that is to say an upper bound which is also known to be a lower bound, under a comparatively mild assumption on the growth behaviour. This extends several statements obtained by Batty, Chill and Tomilov (J. Eur. Math. Soc., vol. 18(4), pp. 853-929, 2016). In fact, for a large class of semigroups our condition is not only sufficient but also necessary for this optimal estimate to hold. Even without this assumption we obtain a new quantified asymptotic result which in many cases of interest gives a sharper estimate for the rate of decay than was previously available, and for semigroups of normal operators we are able to describe the asymptotic behaviour exactly. We illustrate the strength of our theoretical results by using them to obtain sharp estimates on the rate of energy decay for a wave equation subject to viscoelastic damping at the boundary.Comment: 25 pages. To appear in Advances in Mathematic

On the use of mechanical and acoustical excitations for selective heat generation in polymer-bonded energetic materials

To address security issues in both military and civilian settings, there is a pressing need for improved explosives detection technologies suitable for trace vapor detection. In light of the strong dependence of vapor pressure on temperature, trace vapor detection capabilities may be enhanced by selectively heating target materials by external excitation. Moreover, polymer-bonded energetic materials may be particularly susceptible to heating by mechanical or acoustical excitation, due to the high levels of damping and low thermal conductivities of most polymers. In this work, the thermomechanical response of polymer-based energetic composites and methods for acoustical excitation are investigated in order to improve the understanding of the temperature rises induced by applied excitation, and to uncover waveforms which may efficiently transmit excitation energy to generate heat and enhance trace vapor detection capabilities. The heat generation in the binder material of energetic and surrogate systems under harmonic excitation was investigated analytically through the application of a viscoelastic material model. Specifically, structural-scale heating was considered under low-frequency direct mechanical excitation as applied to a beam geometry. Experiments were conducted with a mock mechanical material, wherein the mechanical and thermal responses were recorded by scanning laser Doppler vibrometry and infrared thermography, respectively. Direct comparisons between the model and experimental results demonstrated good agreement with the predicted response, with low-order, bulk-scale heating observed along the modal structure in areas of higher strains. In addition, localized heating near individual crystals was investigated analytically by extending the viscoelastic heating model to general three-dimensional stress-strain states. Application of the model to a Sylgard 184 binder system with an embedded HMX (octogen) crystal under ultrasonic excitation revealed predictions of significant heating rates, particularly near the front edge of the crystal, due to the wave scattering and the resulting stress concentrations. In considering methods for such excitation through incident acoustical or ultrasonic waves, the form of the wave profile was tuned in this work for the purpose of maximizing the energy transmission into solid materials. That transmission is generally limited by the large impedance mismatch at typical fluid--solid interfaces, but by varying the spatial distribution of the incident wave pressure, significant transmission increases can be achieved. In particular, tuned incident inhomogeneous plane waves were found to predict much lower values of the reflection coefficient, and hence large increases in the energy transmission in the context of lossless and low-loss dissipative media. Also, material dissipation was found to have a strong effect on the optimal incident waveform, generally causing a shift to lower inhomogeneity values. Similar results were obtained for parameterized forms of bounded incident waves with respect to the local reflection phenomena and surface wave excitation. These results suggest that, depending on the targeted solid material, substantial energy transmission and heat generation increases may be achieved by tailoring the spatial form of the incident wave profile

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