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Existence of global attractors for the coupled system of suspension bridge equations. (English) Zbl 1080.35008

The authors consider a one-dimensional time-dependent nonlinear system of two partial differential equations \[ \begin{aligned} & u_{tt} + \alpha u_{xxxx} + \gamma_{11} u_{t} + \gamma_{12} u_{txxxx} + k(u-v)^{+} + h_{1}(u, v) = g_{1},\\ & v_{tt} - \beta v_{xx} + \gamma_{21} v_{t} - k (u - v)^{+} + h_{2}(u, v) = g_{2},\\ & u(0, t) = u(L, t) = u_{xx}(0, t) = u_{xx}(L, t) = 0,\qquad t\geq 0,\\ & u(0, t) = u(L, t) = 0, \qquad t\geq 0,\\ & u(x, 0) = u_{0},\quad u_{t}(x, 0) = u_{1},\qquad x\in(0, L),\\ & v(x, 0) = v_{0},\quad v_{t}(x, 0) = v_{1},\qquad x\in(0, L),\\ \end{aligned} \] where \(\gamma_{11},\gamma_{12},\gamma_{21}\geq 0\), \(\alpha > 0\), \(\beta > 0\) and \(g_{1}, g_{2}\in L^{2}(0,L)\). Such a system can represent a one-dimensional nonlinear string-beam system describing the vertical oscillations of a suspension bridge which is coupled with the main cable by the stays. The main cable is modelled as a vibrating string and the roadbed of the bridge is represented by a bending beam with simply supported ends. Using Faedo-Galerkin method combined with a semigroup approach, the authors prove the existence of an absorbing set for the solution of the system. Moreover, the existence of a global attractor of the semigroup associated with the system is obtained.

MSC:

35B41 Attractors
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
35L55 Higher-order hyperbolic systems
35L35 Initial-boundary value problems for higher-order hyperbolic equations
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References:

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