## 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

### Keywords:

nonlinear force; infinite homogeneous string
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