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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:
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
74H20Existence of solutions for dynamical problems in solid mechanics
74H25Uniqueness of solutions for dynamical problems in solid mechanics
74H40Long-time behavior of solutions for dynamical problems in solid mechanics
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References:
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