Vibrations of extensible beams: Unilateral problem

The present work is dedicated to study a unilateral problem relating to the operatorLˆu(x,t)=∂2u∂t2−[aˆ(t)+bˆ(t)∫α(t)β(t)(∂u∂x)2dx]∂2u∂x2+q∂4u∂x4, which models small transverse deflections u(x,t) of an extensible beam with moving ends. Without restriction on the initial configuration u0 and considering the initial velocity u1 with a bounded gradient, we succeed to prove that, given T an arbitrary positive real number, there exists a unique solution for the unilateral problem defined for all t∈[0,T].