Small time-periodic solutions to a nonlinear equation of a vibrating string. (English) Zbl 0653.35063

Feireisl, Eduard

Apl. Mat. 32, 480-490 (1987).

Making use of the accelerated convergence method the author proves existence of at least one \(\omega\)-periodic \((u_ 1,u_ 2)\) to the system \[ u_{1tt}+\sigma_ 1(u_{1x},u_{2x},u_{1xx},u_{1t})=f_ 1,\quad -u_{2xx}+\sigma_ 2(u_{1x},u_{1xx})=f_ 2 \] with Dirichlet boundary conditions where \(f_ j\) are \(\omega\)-periodic in t, small and sufficiently smooth. In a special case, the system above reduces to an integrodifferential equation describing vibrations of a damped extensive string.

Reviewer: O.Vejvoda

MSC:

35L70	Second-order nonlinear hyperbolic equations
35B10	Periodic solutions to PDEs

Keywords:

nonlinear string equation; accelerated convergence; existence; periodic; Dirichlet boundary conditions; vibrations; damped extensive string

References:

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