Non-locality as a regularization mechanism in elastodynamics

It is well known that the solutions to the elastodynamic problem do not satisfy continuous dependence properties on the initial values, and/or supply terms, when the elastic tensor fails to be positive. In fact, the behavior of the solutions can be very explosive since the elements of the spectrum can go to infinite. Therefore, it is very relevant to identify thermomechanical mechanisms regularizing the behavior of the solutions. So, the main aim of this note is to show, from an analytical point of view, how the non-locality, in the sense of Eringen, is a mechanism satisfying this property of regularization of the solutions. It is worth noting that such system has not been previously studied from an analytical point of view. We firstly obtain the existence of the solutions to this problem, even when we do not assume any positivity on the elastic tensor. This result is proved with the help of the linear semigroups theory; however, even with these regularizing effects, the solutions to this problem are unstable. A particular easy one-dimensional problem is also considered. The extension of the existence and instability results to the thermoelastic case is pointed out later. Finally, we also study the spatial behavior of the solutions to the problem in the case that the region is a semi-infinite cylinder, and we obtain a Phragmen–Lindelöf alternative of the exponential type. This result is also relevant because a similar result, without considering regularizing terms, is unknown if the elastic tensor is not positive definite.