We consider the global Cauchy problem for an evolution equation which models an Euler-Bernoulli vibrating beam with time dependent elastic modulus under a force linear function of the displacement u, of the slope u_x, of u_xx and u_xxx. These two last derivatives are proportional to the bending moment and to the shear respectively. We show results of well-posedness in Sobolev spaces assuming that the coefficient of the shear term has a decay rate |x|^(-a), a greater or equal to 1, as x goes to infinity and that all the coefficients of u_x,u_xx and u_xxx satisfy suitable Levi conditions if we allow the elastic modulus to vanish at some time t=t_0.
The global Cauchy problem for a vibrating beam equation
ASCANELLI, Alessia;
2009
Abstract
We consider the global Cauchy problem for an evolution equation which models an Euler-Bernoulli vibrating beam with time dependent elastic modulus under a force linear function of the displacement u, of the slope u_x, of u_xx and u_xxx. These two last derivatives are proportional to the bending moment and to the shear respectively. We show results of well-posedness in Sobolev spaces assuming that the coefficient of the shear term has a decay rate |x|^(-a), a greater or equal to 1, as x goes to infinity and that all the coefficients of u_x,u_xx and u_xxx satisfy suitable Levi conditions if we allow the elastic modulus to vanish at some time t=t_0.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
