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 equation u '' + A(2)u M(\\A(1/2)u\\(2)(H))Au + g(u') = 0, suggested by the study of plates and beams, where A is a linear operator in a Hilbert space H and A and g are real functions. We also study the asymptotic behaviour of the solutions, under suitable growth assumptions on the nonlinear term g. (C) 1997 Academic Press.