PROPERTY OF GROWTH DETERMINED BY SPECTRUM OF OPERATOR ASSOCIATED WITH THE TIMOSHENKO SYSTEM WITH WEAKLY DISSIPATION

In this work, we consider the Timoshenko system with weakly dissipation, one dissipation, phi(t), on the transverse displacement and another psi(t), on the rotation angle of a filament of the beam rho(1)phi(tt) - kappa(phi(x) + psi)(x) + phi(t) = 0, in (0,L) x (0,t), rho(2)psi(tt) - b psi(xx) + kappa(phi(x) + psi) + psi(t) = 0, in (0,L) x (0,t), with initial conditions phi(x,0) = phi(0)(x), phi(t)(x,0) = phi(1)(x), psi(x,0) = psi(0)(x), psi(t)(x,0) = psi(1)(x). In [13] was proved the exponential stability of the semigroup associated with this system, and now we prove the property of growth determined by spectrum of operator associated, present the type of semigroup and also indicate the best constant for the exponential stability.