## 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 $$\mathbb R^{n+1}$$ $$(n\geq1)$$ 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 $$n\geq 1$$.

### MSC:

 35L75 Higher-order nonlinear hyperbolic equations 35L35 Initial-boundary value problems for higher-order hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs 45K05 Integro-partial differential equations

### Keywords:

exponential decay; noncylindrical domain; regular solutions
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