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On a beam equation in Banach spaces. (English) Zbl 1399.35259
Summary: This paper is concerned with the existence and asymptotic behavior of solutions of the Cauchy problem for an abstract model for vertical vibrations of a viscous beam in Banach spaces. First is obtained a local solution of the problem by using the method of successive approximations, a characterization of the derivative of the nonlinear term of the equation defined in a Banach space and the Ascoli-Arzela theorem. Then the global solution is found by the method of prolongation of solutions. The exponential decay of solutions is derived by considering a Lyapunov functional.
MSC:
35L05 Wave equation
35B40 Asymptotic behavior of solutions to PDEs
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