Measure-Dependent Stochastic Nonlinear Beam Equations Driven by Fractional Brownian Motion

We study a class of nonlinear stochastic partial differential equations arising in the mathematical modeling of the transverse motion of an extensible beam in the plane. Nonlinear forcing terms of functional-type and those dependent upon a family of probability measures are incorporated into the initial-boundary value problem (IBVP), and noise is incorporated into the mathematical description of the phenomenon via a fractional Brownian motion process. The IBVP is subsequently reformulated as an abstract second-order stochastic evolution equation driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space and is studied using the tools of cosine function theory, stochastic analysis, and fixed-point theory. Global existence and uniqueness results for mild solutions, continuous dependence estimates, and various approximation results are established and applied in the context of the model. 1. Introduction The mathematical description of the dynamic buckling has been the subject of investigation for decades and is of interest to the engineering world. Dynamic buckling arises in various ways including vibrations, single load pulses of large amplitude, occurrence of a suddenly applied load, and flutter enhanced bending (see [1]). For the purpose of our study we restrict our attention to the dynamic buckling of a hinged extensible beam which is stretched or compressed by an axial force. Dickey [2] initiated an investigation of the hyperbolic partial differential equation where is Young’s modulus, is the cross-sectional moment of inertia, is the density, is an axial force,？？ is the natural length of the beam, and is the cross-sectional area; these are all positive parameters. Here, describes the transverse deflection of an extensible beam at point at time . The nonlinear term in (1) accounts for the change in tension of the beam due to extensibility. The ends of the beam are held at a fixed distance apart and the ends are hinged; this translates to the following boundary conditions: The initial conditions given by describe the initial deflection and initial velocity at each point of the beam. Fitzgibbon [3] and Ball [4] established the general existence theory for IBVP (1)–(3). Patcheu [5] subsequently incorporated a nonlinear friction force term into the mathematical model to account for dissipation; this was done by replacing the right side of (1) by the term , where is a bounded linear operator. More recently, Balachandran and Park [6] further introduced the term into (1) to account for the