Existence and uniform decay for a nonlinear beam equation with nonlinearity of Kirchhoff type in domains with moving boundary. (English) Zbl 1092.35068

Santos, M. L.; Ferreira, J.; Raposo, C. A.

Abstr. Appl. Anal. 2005, No. 8, 901-919 (2005).

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