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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.

##### MSC:
 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
##### Keywords:
existence; uniqueness; Faedo-Galerkin method; special basis