Existence and uniform decay for a nonlinear beam equation with nonlinearity of Kirchhoff type in domains with moving boundary

We prove the exponential decay in the case n > 2 , as time goes to infinity, of regular solutions for the nonlinear beam equation with memory and weak damping u t t + Δ 2 u − M ( ‖ ∇ u ‖ L 2 ( Ω t ) 2 ) Δ u + ∫ 0 t g ( t − s ) Δ u ( s ) d s + α u t = 0 in Q ^ in a noncylindrical domain of ℝ n + 1 ( n ≥ 1 ) under suitable hypothesis on the scalar functions M and g , and where α is a positive constant. We establish existence and uniqueness of regular solutions for any n ≥ 1 .