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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.

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