Longtime Behavior For The Generalized Damped Extensible Beam Equation

The paper studies the longtime behavior of solutions to the initial boundary value problem(IBVP) for a class of nonlinear wave equation with weak dampingA mathematical model for the problem is one-dimentional beam equationwhich was introduced by Woinowsky-Kriger [5] withλ= g= f = 0asa model for the transversemotion of an extensible beam whose ends are held a fixed distance apart. By virtue of operatortheory, we get the Cauchy problem which equivalents the original one as followswhere A=△²,D(A)=H⁴∩H₀². We prove the global existence and uniqueness of solutions inC(R⁺;D(A^1/2))∩C¹(R⁺;H) by Galerkin method. And the global solution is regularized in spaceC(R⁺;D(A^3/4))∩C(R⁺;D(A^1/4)). Then C₀-semigroups are defined in different spaces. It provesthat the related continuous semigroup possesses in the different phase spaces a global attractorwhich is connected. And an example is shown.