Spatial behavior for a fourth-order dispersive equation

Abstract

Abstract. In this note we investigate the spatial behavior of a linear equation of fourth order which models several mechanical situations when dispersive and dissipative effects are taken into account. In particular, this equation models the extensional vibration of a bar when we assume that external friction, with a rough substrate for example, is present. We show that for such an equation a Phragmén-Lindelöf alternative of exponential type can be obtained. A bound for the amplitude term in terms of boundary data is obtained. Moreover, when friction is absent, we obtain exponential decay results in the case of harmonic vibrations and we prove a polynomial decay estimate for general solutions. Introduction.In recent years much attention has been directed to the study of the damping of end effects in several thermomechanical situations and to the general study of the spatial behavior of solutions of partial differential equations and systems. The history and development of these questions was extensively surveyed by Horgan and Knowles [11] in a work periodically updated by Horgan [9,10]. Moreover, the books by Ames and Straughan [1], Flavin and Rionero [6] and Antontsev et al. [2] are devoted to such matters. The energy method is widely used to study the spatial behavior of solutions of partial differential systems.In this note we investigate the spatial behavior of a linear equation of fourth order which is encountered in several frameworks when we take into account dispersive effects.

Phragmén–Lindelöf alternative of exponential type for the solutions of a fourth order dispersive equation

“…The present note is a complement to the paper [7]. Here by using the relation (2.4) we are able to control the term u ;t .…”

“…We note that we had the term ðu ;t À esu ;t Þ 2 in (2.3) and then we could not control the time derivative of the solution. In [7] only a polynomial decay result has been provided. It is worth noting that we also need to control 'u ;t , but we also have this in the RHS of (2.4).…”

“…For these reasons in [7], we investigated the spatial behavior of the dispersive fourth-order linear equation u ;tt À esu ;tt ¼ asu: ð1:2Þ…”

“…In [7] the exponential decay was established also for the harmonic vibration problem in a cylindrical domain. It is interesting therefore to improve the result in [7] about the polynomial decay estimate for the whole class of solutions of (1.2). The aim of the present note is exactly to show a Phragmén-Lindelö f alternative of exponential type for the solutions of this equation.…”

Polynomial and Exponential Spatial Decay Estimates in Elasticity

“…This argument is very recent in the study of spatial estimates and, to my knowledge, has been used only three times in the literature [12][13][14].…”