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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

u tt +αu xxxx +γ 11 u t +γ 12 u txxxx +k(u-v) + +h 1 (u,v)=g 1 ,v tt -βv xx +γ 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,t0,u(0,t)=u(L,t)=0,t0,u(x,0)=u 0 ,u t (x,0)=u 1 ,x(0,L),v(x,0)=v 0 ,v t (x,0)=v 1 ,x(0,L),

where γ 11 ,γ 12 ,γ 21 0, α>0, β>0 and g 1 ,g 2 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