Stochastic nonlinear beam equations
The nonlinear beam equation \[ {\partial^2 y\over\partial t^2}+ y{\partial^2 y\over \partial x^4}= \Biggl[a+ b\int^l_0 \Biggl({\partial y\over\partial x}\Biggr)^2 dx\Biggr] {\partial^2y\over\partial x^2} \] was proposed by \textit{S. Woinowsky-Krieger} [J. Appl. Mech. 17, 35--36 (1950; Zbl 0036.13302)] as a model for the transversal deflection of an extensible beam of natural length \(l\), having the ends fixed at the support, under an axial force. Motivated by problems arising in aeroelasticity, \textit{P. L. Chow} and \textit{J. L. Menaldi} [Differ. Integral Equ. 12, No.~3, 419--434 (1999; Zbl 1006.60058)] considered a beam described by the equation and subjected to a force including random fluctuations. The stability conditions proposed in the present paper utilize in a substantial way the specific form of the problem \[ u_{tt}+ A^2u+ g(u,u_t)+ m(|B^{1/2}|^2)Bu= \sigma(u,u_t)W \] in a Hilbert space \(H\), where the operators \(A\) and \(B\) are positive selfadjoint and \(W\) is an infinite-dimensional Wiener process. In the next section, the problem is introduced precisely and main results are stated, their proofs are being deferred to Sections 2, Existence of global solutions, and 3, Stability of the zero solution. In Section 4, the authors show that the abstract model and study covers particular problems, and in Appendix they provide a lemma justifying the definition of a local mild solution they use.
scientific article; zbMATH DE number 2164756
scientific article