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Existence and uniform decay for a nonlinear beam equation with nonlinearity of Kirchhoff type in domains with moving boundary. (English) Zbl 1092.35068
Summary: We prove the exponential decay in the case $n>2$, as time goes to infinity, of regular solutions for the nonlinear beam equation with memory and weak damping ${u}_{tt}+{{\Delta }}^{2}u-{M\left(\parallel \nabla u\parallel }_{{L}^{2}\left({{\Omega }}_{t}\right)}^{2}\right){\Delta }u+{\int }_{0}^{t}g\left(t-s\right){\Delta }u\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds+\alpha {u}_{t}=0$ in $\stackrel{^}{Q}$ in a noncylindrical domain of ${ℝ}^{n+1}$ $\left(n\ge 1\right)$ under suitable hypothesis on the scalar functions $M$ and $g$, and where $\alpha$ is a positive constant. We establish existence and uniqueness of regular solutions for any $n\ge 1$.
##### MSC:
 35L75 Nonlinear hyperbolic PDE of higher $\left(>2\right)$ order 35L35 Higher order hyperbolic equations, boundary value problems 35B40 Asymptotic behavior of solutions of PDE 45K05 Integro-partial differential equations