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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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