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On the solutions for an extensible beam equation with internal damping and source terms. (English) Zbl 1439.35073

Summary: In this manuscript, we consider the nonlinear beam equation with internal damping and source term \[u_{tt}+\Delta^2 u+M(| \nabla u|^2)(- \Delta u) +u_t=|u|^{r-1} -u \] where \(r>1\) is a constant, \(M(s)\) is a continuous function on \([0,+ \infty)\). The global solutions are constructed by using the Faedo-Galerkin approximations, taking into account that the initial data is in appropriate set of stability created from the Nehari manifold. The asymptotic behavior is obtained by the Nakao method.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L35 Initial-boundary value problems for higher-order hyperbolic equations
35L76 Higher-order semilinear hyperbolic equations
74K20 Plates
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
93D15 Stabilization of systems by feedback
93D20 Asymptotic stability in control theory
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