Undamped nonlinear beam excited by additive L2L2-regular noise

Stability of a nonlinear elastic beam excited by a space-time dependent nonrandom force or a L2L2-regular random noise f(x,t)f(x,t) (white in time, with general spatial covariance) is considered. It is supposed that the transversal displacement yy of the beam with length ll is governed by the quasilinear partial differential equation (PDE) ρ0hytt+Dyxxxx−[N0+ε∫0l(yx′)2dx′]yxx+ky=f(x,t),0