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Existence, uniqueness and uniform decay for the nonlinear beam degenerate equation with weak damping. (English) Zbl 1063.74067
Summary: We prove global existence and uniqueness of weak solutions of the problem for the nonlinear beam degenerate equation \[ K(x, t)u''+ \Delta^2u+ M\Biggl(\int_\Omega |\nabla u|^2 dx\Biggr)(-\Delta u)+ u'= 0\quad\text{in }Q= \Omega\times (0,\infty), \] where \(Q\) is a cylindrical domain of \(\mathbb{R}^{n+1}\), \(n\geq 1\), with the lateral boundary \(\Sigma\), and \(K(x,t)\) is a real function defined in \(Q\), \(K(x,t)\geq 0\) for all \((x,t)\in\Omega\times (0,\infty)\), which satisfies some appropriate conditions. \(M(\lambda)\) is a real function such that \(M(\lambda)\geq -\beta\), \(0< \beta< \lambda_1\), \(\lambda_1\) is the first eigenvalue of the operator \(\Delta^2\). Moreover, the uniform decay rates of the energy are obtained as time goes to infinity.

MSC:
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H20 Existence of solutions of dynamical problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
74H40 Long-time behavior of solutions for dynamical problems in solid mechanics
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