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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 $\bbfR^{n+1}$, $n\ge 1$, with the lateral boundary $\Sigma$, and $K(x,t)$ is a real function defined in $Q$, $K(x,t)\ge 0$ for all $(x,t)\in\Omega\times (0,\infty)$, which satisfies some appropriate conditions. $M(\lambda)$ is a real function such that $M(\lambda)\ge -\beta$, $0< \beta< \lambda_1$, $\lambda_1$ 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.) in solid mechanics 74H20 Existence of solutions for dynamical problems in solid mechanics 74H25 Uniqueness of solutions for dynamical problems in solid mechanics 74H40 Long-time behavior of solutions for dynamical problems in solid mechanics
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