Longtime Behavior For The Beam Equation With Weak Damping

This paper studies the longtime behavior of solutions to a class beam equation with weak dampingEspecially,when the dimension N =1,a mathematical model for the problem is onedimentiondal beam equationwhich was introduced by Woionwsky-Kriger[l] withλ=g =f = 0asa model for the transvers motion of an extensible beam whose are held. Were write the (1)-(3) as the abstract Cauchy problemwhere A=Δ²,D(A)=H⁴∩H₀². We first established the global existence and uniqueness of solution of the Cauchy problem in C(R⁺;D(A^?)∩C(R⁺;D(A^δ/4)),(0≤δ≤1),then we prove that the related continuous semigroup possesses a global attractor in the phase space E₁=D(A^?)×D(A^δ/4).Finally,we have an example.