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On the asymptotic behavior for a damped oscillator under a sublinear friction. (English) Zbl 1018.34051

Summary: The authors show that there are two curves of initial data \((x_0, v_0)\) for which the solution \(x(t)\) to the corresponding Cauchy problem associated to the equation \(x_{tt}+|x_t|^{\alpha-1} x_t+ x=0\), with \(\alpha\in (0,1)\), vanishes after a finite time. By using asymptotic methods and comparison arguments, the authors show that for many other initial data the solutions decay to \(0\), in an infinite time, as \(t^{-\alpha/(1-\alpha)}\).

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
70K20 Stability for nonlinear problems in mechanics
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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