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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.

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
Full Text: DOI
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