On a Global Solution and Asymptotic Behaviour for the Generalized Damped Extensible Beam Equation

We prove the existence and uniqueness of global solutions for the Cauchy problem concerning the evolution equationu″+A2u+M( A1/2u 2H) Au+g(u′)=0,suggested by the study of plates and beams, whereAis a linear operator in a Hilbert spaceHandMandgare real functions. We also study the asymptotic behaviour of the solutions, under suitable growth assumptions on the nonlinear termg