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A monotonicity method for solving hyperbolic problems with hysteresis. (English) Zbl 0668.35065

A method of Minty-Browder type is used for proving the existence and uniqueness of a weak \(\omega\)-periodic solution to the model equation for vibrating processes in elasto-plastic solids or in ferromagnetics \(u_{tt}-div(F(\text{grad} u))=g\) in a bounded domain \(\Omega \subset R^ N\), \(u=0\) on \(\partial \Omega\), where g is given \(\omega\)-periodic function and F is the Ishlinskij hysteresis operator. Its hyperbolicity is confirmed by the finite speed of propagation. The proof is based on sharp estimates of hysteresis energy losses.
Reviewer: P.Krejčí

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B10 Periodic solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
74H99 Dynamical problems in solid mechanics
74H45 Vibrations in dynamical problems in solid mechanics
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References:

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