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Vibrations of Elastic Strings: Mathematical Aspects, Part Two

Authors:

Institution:

(1) Instituto de Matemática UFRJ, Rio de Janeiro, RJ, Brasil. E-mail;(2) Instituto de Matemática UFF, Niterói, RJ, Brasil;(3) Departamento de Matemática UFPA, Belém, PA, Brasil

Abstract:

Dedicated to Professor Jacque-Louis Lions on the occasion of his 70th birthday We consider a mixed problem for the operator

$\hat Lu\left( {x,t} \right) = \frac{{\partial ^2 u}}{{\partial t^2 }} - \left( {a\left( t \right) + b\left( t \right)\int_{\alpha \left( t \right)}^{\beta \left( t \right)} {\left( {\frac{{\partial u}}{{\partial x}}} \right)} ^2 dx} \right)\frac{{\partial ^2 u}}{{\partial x^2 }}$

in a noncylindrical domain $\widehat Q$ . We obtain local solution in t. When we add a viscosity we obtain a global solution. We also investigate the asymptotic behavior of the energy.

Keywords:

Elastic strings viscosity asymptotic behavior

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