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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 $u_{tt}+ \Delta^2u- M(\|\nabla u\|_{L^2(\Omega_t)}^2) \Delta u+\int_0^t g(t-s) \Delta u(s)\,ds+ \alpha u_t=0$ in $\widehat{Q}$ in a noncylindrical domain of $\Bbb R^{n+1}$ $(n\ge1)$ 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 $n\ge 1$.

35L75Nonlinear hyperbolic PDE of higher $(>2)$ order
35L35Higher order hyperbolic equations, boundary value problems
35B40Asymptotic behavior of solutions of PDE
45K05Integro-partial differential equations
Full Text: DOI EuDML