Existence, uniqueness and uniform decay for the nonlinear beam degenerate equation with weak damping

In this paper we prove global existence and uniqueness of weak solutions of the problem for the nonlinear beam degenerate equationK(x,t)u^''[email protected]^[email protected]!"@W|@?u|^2dx([email protected])+u^'[email protected](0,~),whereQ is a cylindrical domain of R^n^+^1, n>=1, with the lateral boundary @S and K(x,t) is a real function defined in Q, K(x,t)>=0 for all (x,t)@[email protected](0,~) which satisfies some appropriate conditions. M(@l) is a real function such that M(@l)>[email protected], 0<@b<@l"1, @l"1 is the first eigenvalue of the operator @D^2. Moreover, the uniform decay rates of the energy are obtained as time goes to infinity.