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

The behaviour of the total energy for a semi-infinite piezoelectric beam, with uniform or variable cross-section, in dynamical conditions is investigated. Precisely, we obtain some estimates in terms of the data for the energy portion of the body at distance greater than z form the base and for its norm in L^1(0,t) (where t is an arbitrary positive time). These estimates depend on the initial data if t is less than or equal to z/V (V = computale positive material constant); if t > z/V, by using the Korn inequality, we show tha the bounds depend on all data. Under mild hypotheses on the initial data we derive the asymptotic behaviour of the energy as z tends to infinity. All possible combinations of boundary conditions are examined and the kind of the estimate is formally the same for all problems whether the beam is a cylinder or not