Stochastic PDE for nonlinear vibration of elastic panels
The paper aims at proving existence and uniqueness of solutions to a nonlinear stochastic hyperbolic equation with a nonlocal term \[ \partial ^2_{t} u - \Biggl [\alpha + \beta \int ^{l}_{0} |\partial _{y}u(t,y)|^2 dy\Biggr ]\partial ^2_{x} u + \gamma \partial ^4_{x} u = F(t,x,\partial _{t}u,\partial _{x}u), \tag{1} \] \(x\in [0,l]\), \(t\geq 0\), with boundary conditions \(u(t,0)=u(t,l) =0\), \(\partial _{x}u(t,0) = \partial _{x}u(t,l) = 0\) and initial conditions \(u(0,\cdot)=\varphi _0\), \(\partial _{t}u(0,\cdot) = \varphi _1\). The constants \(\alpha \), \(\beta \), \(\gamma \) are positive, \(F\) is a random force, \[ F(t,x,r,v)= f(t,x) + \sum ^\infty _{k=1} \sigma _{k}(t,x)\dot w_{k}(t) + g(t,x,r,v) + \sum ^{\infty } _{k=1} \rho _{k}(t,x,r,v)\dot B_{k}(t), \] where \(w_{k}\), \(B_{k}\) are independent standard one-dimensional Wiener processes. Equations of this type arise in nonlinear aeroelasticity; existence and uniqueness of local solutions (up to an explosion time \(\tau \)) follow from standard theory; however, it is not clear at all whether \(\tau =\infty \). In the paper, global solutions are constructed under suitable Lipschitz continuity and linear growth hypotheses upon the functions \(g\), \(\sigma _{k}\), \(\rho _{k}\).
stochastic partial differential equations
scientific article; zbMATH DE number 1858754