Here, is the time variable and x is the space coordinate along the beam in its equilibrium position. We denote by w(x, t) the deflection of the beam from the equilibrium line, which is described by w 0, and by (x, t) the slope of the deflection cue when the shearing force is neglected; for the precise meaning of , see Timoshenko 11 or Traill-Nash and Collar 12]. We assume that the motion occurs in the wx-plane and that 0 x L. The coecients p, I, E and I are the mass per unit length, the mass moment of ineia of the cross section, Young's modulus and the moment of ineia of the cross section, respectively. The coecient K is equal to kGA, where G is the modulus of elasticity in shear, A is the cross sectional area and k is a numerical factor depending on the shape of the cross section. The boundary condition we employ
In this paper, we discuss an initial-boundary value problem and a Cauchy problem for the stochastic porous medium equation. Our basic estimates are based on the known results due to (Math. Sbornik 67 (1965) 609-642, Arch. Rational Mech. Anal. 25 (1967) 64-80). By the procedure developed in (Trans. Amer. Math. Soc. 354 (2001) 1117-1135), we obtain solutions over the given probability space rather than martingale solutions. We will also establish the existence of invariant measures when the space domain is bounded.
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