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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 $$ \align & u_{tt} + \alpha u_{xxxx} + \gamma_{11} u_{t} + \gamma_{12} u_{txxxx} + k(u-v)^{+} + h_{1}(u, v) = g_{1},\\ & v_{tt} - \beta v_{xx} + \gamma_{21} v_{t} - k (u - v)^{+} + h_{2}(u, v) = g_{2},\\ & u(0, t) = u(L, t) = u_{xx}(0, t) = u_{xx}(L, t) = 0,\qquad t\ge 0,\\ & u(0, t) = u(L, t) = 0, \qquad t\ge 0,\\ & u(x, 0) = u_{0},\quad u_{t}(x, 0) = u_{1},\qquad x\in(0, L),\\ & v(x, 0) = v_{0},\quad v_{t}(x, 0) = v_{1},\qquad x\in(0, L),\\ \endalign $$ where $\gamma_{11},\gamma_{12},\gamma_{21}\ge 0$, $\alpha > 0$, $\beta > 0$ 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:
35B41Attractors (PDE)
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
37L05General theory, nonlinear semigroups, evolution equations
35L55Higher order hyperbolic systems
35L35Higher order hyperbolic equations, boundary value problems
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References:
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