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Instability of localized buckling modes in a one-dimensional strut model. (English) Zbl 0895.35102
The paper concerns the stability of equilibrium solutions \(y\) of the equation \[ u_{tt}+ u_{xxxx} +Pu_{xx} +u-u^2 =0 \] (representing a mechanical strut on an elastic foundation with nonlinear restoring force) which are local in the sense that \(u(x)\to 0\) as \(x\to \pm \infty\). The primary buckling mode is shown to be unstable by considering the second variation \[ L(h)v =v_{xxxx} +Pv_{xx} +(1-2h)v \] and showing that \(L(h)\) has at least one negative eigenvalue. Stability in the case of rigid loading is proved when the solution is transversely constructed (i.e. stable and unstable manifolds intersect transversely at \(h(0)\) in the zero set of the energy) and the energy is strictly decreasing near \(P\). A similar analysis is carried out for multimodal solutions.

35Q72 Other PDE from mechanics (MSC2000)
74B20 Nonlinear elasticity
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