Spatial energy estimates in dynamical problems for a semi-infinite piezoelectric beam

The behaviour of the total energy for a semi-infinite piezoelectricbeam, with uniform or variable cross-section, in dynamical conditionsis investigated. Precisely, we obtain some estimates in termsof the data for the energy of the portion of the body at distancegreater than z from the base and for its norm in L¹(0,t) (wheret is an arbitrary positive time). We obtain some estimates whichdepend on the initial data if t z/V (where V is a computablepositive material constant); if t > z/V, by using also theKorn inequality, we show that the bounds depend on all the data.Under mild hypotheses on the initial data we derive the asymptoticbehaviour of the energy as z +. All possible combinations ofboundary conditions are examined and the kind of the estimateis formally the same for all the problems whether the beam isa cylinder or not.