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Vibrations of beams by torsion or impact (mathematical analysis). (English) Zbl 1195.35045
Summary: This article contains a mathematical analysis of the initial boundary value problem
\[ \begin{cases} u''(x,t)-\Delta u(x,t)+\delta(x)u'(x,t)=0 &\text{in }\Omega\times(0,\infty),\\ u=0 &\text{on }\Gamma_0\times(0,\infty,\\ \frac{\partial u}{\partial\nu}+ \alpha(x)u''(x,t)+ \beta(x)u'(x,t)=0 &\text{on }\Gamma_1\times(0,\infty,\\ u(x,0)= u^0(x),\quad u'(x,0)=u^1(x) &\text{in }\Omega. \end{cases} \]
It was motivated by a torsion or impact of cylindrical beams. With restrictions on \(\delta\), \(\alpha\), \(\beta\), \(u^0\), \(u^1\) we prove the existence and uniqueness of solutions and asymptotic behavior of the energy. We employ Faedo-Galerkin method with a special basis.

35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35L05 Wave equation
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35L20 Initial-boundary value problems for second-order hyperbolic equations