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: rho partial derivative(2)(t)u + delta partial derivative(t)u + kappa partial derivative(4)(x)u + eta partial derivative(t)partial derivative(4)(x)u = (alpha + beta integral(R) vertical bar partial derivative(x)u vertical bar(2)dx + gamma eta integral partial derivative(t)partial derivative(x)u partial derivative(x)udx) partial derivative(2)(x)u. The aim of this paper is to investigate singular limits as rho -> 0 for this problem. In the authors' previous paper [8] decay estimates of solutions u(rho) to the equation in the case rho > 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: parallel to u(rho) - u(0)parallel to(L infinity ([0,infinity);H2(R))) <= C rho. (C) 2015 Elsevier Inc. All rights reserved.