Singular limits in the Cauchy problem for the damped extensible beam equation

We study the Cauchy problem of the Ball model for an extensible beam:ρ∂t2u+δ∂tu+κ∂x4u+η∂t∂x4u=(α+β∫R|∂xu|2dx+γη∫R∂t∂xu∂xudx)∂x2u. The aim of this paper is to investigate singular limits as ρ→0ρ→0 for this problem. In the authors' previous paper [8] decay estimates of solutions uρuρ to the equation in the case ρ>0ρ>0 were shown. With the help of the decay estimates we describe the singular limit in the sense of the following uniform (in time) estimate:‖uρ−u0‖L∞([0,∞);H2(R))≤Cρ.‖uρ−u0‖L∞([0,∞);H2(R))≤Cρ.