ON THE ASYMPTOTICS OF SOME STRONGLY DAMPED BEAM EQUATIONS WITH STRUCTURAL DAMPING

The Fourier transform, F, on R-N (N >= 1) transforms the Cauchy problem for a strongly damped beam equation with structural damping u(tt) - delta u(t) + alpha(delta(2))u - delta u = 0, alpha >= 0, to an ordinary differential equation in time. With u(t, x) being the weak solution of the problem given by the Fourier transform, the goal of the paper is to determine the asymptotic expansion of the squared L-2-norm of u(t, x) as t -> infinity. With suitable, additional assumptions on the initial data u(0, x) and u(t)(0, x), we establish the behavior of the squared L-2-norm of u(t, x) as t -> infinity.