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Existence and boundary stabilization of solutions for the Kirchhoff equation. (English) Zbl 0930.35110
Summary: This paper is concerned with the existence of local and global solutions of an initial-homogeneous boundary value problem for the Kirchhoff equation \[ u''- M\Biggl(t, \int_\Omega|\nabla u|^2 dx\Biggr) \Delta u= 0, \] where \(M(t,\lambda)\geq m_0> 0\) and \(\Omega\) is an open bounded set of \(\mathbb{R}^n\). The boundary stability is also obtained. The fixed point method, Galerkin approximations and energy functionals are used in the approach.

MSC:
35L70 Second-order nonlinear hyperbolic equations
93D15 Stabilization of systems by feedback
35R10 Functional partial differential equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
74K05 Strings
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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