## 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
Full Text:

### References:

 [1] Lazer, A. C.; McKenna, P. J., Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Rev., 32, 537-578 (1990) · Zbl 0725.73057 [2] Ahmed, N. U.; Harbi, H., Mathematical analysis of dynamic models of suspension bridges, SIAM J. Appl. Math., 58, 853-874 (1998) · Zbl 0912.93048 [3] Humphreys, L. D., Numerical mountain pass solutions of a suspension bridge equation, Nonlinear Anal., 28, 1811-1826 (1997) · Zbl 0877.35126 [5] Vejvoda, O., Partial Differential Equations: Time-Periodic Solutions (1982), Nijhoff: Nijhoff Hague · Zbl 0183.10401 [6] McKenna, P. J.; Walter, W., Nonlinear oscillation in a suspension bridge, Arch. Rational Mech. Anal.. Arch. Rational Mech. Anal., Nonlinear Anal., 39, 731-743 (2000), Results [7] Lazer, A. C.; McKenna, P. J., Large scale oscillatory behavior in loaded asymmetric systems, Anal. Nonlinear I.H.P., 4, 243-274 (1987) · Zbl 0633.34037 [8] Giorgi, C.; Muñoz Rivera, J. E.; Pata, V., Global attractors for a semilinear hyperbolic equations in viscoelasticity, J. Math. Anal. Appl., 260, 83-99 (2001) · Zbl 0982.35021 [9] Pata, V.; Zucchi, A., Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11, 505-529 (2001) · Zbl 0999.35014 [10] Ma, Q. F.; Wang, S. H.; Zhong, C. K., Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J., 51, 1541-1559 (2002) · Zbl 1028.37047 [11] Teman, R., Infinite Dimensional Dynamical System in Mechanics and Physics (1997), Springer-Verlag: Springer-Verlag Berlin · Zbl 0871.35001 [12] Ma, Q.; Zhong, C., Existence of strong global attractors for hyperbolic equation with linear memory, Appl. Math. Comput., 157, 745-758 (2004) · Zbl 1068.45018 [13] Ma, Q.; Zhong, C., Global attractors of strong solutions for nonclassical diffusion equation, J. Lanzhou Univ., 40, 7-9 (2004) · Zbl 1090.35145
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.