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Existence of strong solutions and global attractors for the coupled suspension bridge equations. (English) Zbl 1176.35036

The authors consider the vibrating beam equation coupled with a vibrating string equation:

$\begin{array}{cc}\hfill {u}_{tt}+\alpha {u}_{xxxx}+{\delta }_{1}{u}_{t}+k{\left(u-v\right)}^{+}+{f}_{B}\left(u\right)={h}_{B}\phantom{\rule{1.em}{0ex}}& \text{in}\phantom{\rule{4.pt}{0ex}}\left[0,L\right]×{ℝ}^{+},\hfill \\ \hfill {v}_{tt}-\beta {v}_{xx}+{\delta }_{2}{v}_{t}-k{\left(u-v\right)}^{+}+{f}_{S}\left(v\right)={h}_{S}\phantom{\rule{1.em}{0ex}}& \text{in}\phantom{\rule{4.pt}{0ex}}\left[0,L\right]×{ℝ}^{+}\hfill \end{array}$

with the simply supported boundary conditions at both ends and initial value conditions. For proper $k$ the existence of the strong solution is obtained by the Faedo-Galerkin method. A priori estimates are considered. The authors prove that the solution semigroup defined on the associated product space has a global attractor.

##### MSC:
 35B41 Attractors (PDE) 35M31 Initial value problems for systems of mixed type 35Q70 PDEs in connection with mechanics of particles and systems 47D06 One-parameter semigroups and linear evolution equations
##### References:
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