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On the solvability of a system of wave and beam equations. (English) Zbl 1006.47054

The author considers a system of wave and beam equations with linear coupling and damping which provides a model for a suspension bridge where the main cable is thought of as a vibrating string and the road bed as a vibrating beam. This system is transformed into an operator equation \(Lu-Au-N(u)=h\) in \(H:=L^2(\Omega)\times L^2(\Omega)\) with \(\Omega=(0,2\pi)\times(0,\pi)\), \(L\) is the differential operator in \(H\) corresponding to the wave operator \(u\mapsto\frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}\) and the beam operator \(\frac{\partial^2v}{\partial t^2}+\frac{\partial^4v}{\partial x^4}+\beta\frac{\partial v}{\partial t}\), \(A\) is the multiplication operator corresponding to the linear coupling, and \(N\) is the Nemytskij operator corresponding to the nonlinearity. The matrix spectrum \(\sigma_M(L)\) of \(L\) is defined to be the set of those \(2\times 2\) real matrices \(B\) such that \(L-B\) is not continuously invertible (where we identify a matrix with the associated multiplication operator on \(H\)). The author here deals with the resonance case \(A\in\sigma_M(L)\), and he is looking for weak solutions. To this end, he imposes conditions on the nonlinearity which enable him to apply the degree theory developed by the author and V. Mustonen [Differ. Int. Equ. 3, No. 5, 945-963 (1990; Zbl 0724.47024)].

MSC:

47N20 Applications of operator theory to differential and integral equations
47H11 Degree theory for nonlinear operators
47J05 Equations involving nonlinear operators (general)
35B10 Periodic solutions to PDEs
35Q72 Other PDE from mechanics (MSC2000)
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)

Citations:

Zbl 0724.47024
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