Existence and uniform decay for a nonlinear beam equation with nonlinearity of Kirchhoff type in domains with moving boundary

Abstract

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({\Vert \nabla u\Vert }_{L^2\left({\Omega }_t\right)}^2\right)\Delta u+{\displaystyle {\int }_0^{t}g}\left(t-s\right)\Delta u\left(s\right)ds+\alpha u_t=0\text{\hspace{0.17em}in\hspace{0.17em}}\widehat{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 $n\ge 1$.