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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: \[ \begin{aligned} u_{tt}+ \alpha u_{xxxx}+\delta_1 u_t+ k(u- v)^++ f_B(u)= h_B\quad &\text{in }[0,L]\times \mathbb{R}^+,\\ v_{tt}- \beta v_{xx}+ \delta_2 v_t- k(u- v)^++ f_S(v)= h_S\quad &\text{in }[0,L]\times \mathbb{R}^+\end{aligned} \] 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
35M31 Initial value problems for mixed-type systems of PDEs
35Q70 PDEs in connection with mechanics of particles and systems of particles
47D06 One-parameter semigroups and linear evolution equations
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