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On stabilization of string-nonlinear oscillator interaction. (English) Zbl 0859.35076

The paper deals with an infinite homogeneous string with a particle of mass \(m\geq 0\) attached at the origin. The particle is subjected to a force \(F=F(u)\) which depends, in general nonlinearly, upon the transversal displacement \(u\) of the particle. Solvability of the Cauchy problem for the system is discussed. Convergence, as time \(t\to\infty\), to a stationary state related to a zero \(b_+\) of \(F\) is proved. Finally, an interesting study is made of the transition from one state \(u=b_-\) to another \(u=b_+\), where \(F(b_{\pm})=0\). With suitable assumptions upon the data such a transition is shown to be always possible.

MSC:

35L70 Second-order nonlinear hyperbolic equations
74K05 Strings
74H45 Vibrations in dynamical problems in solid mechanics
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