Lin, Jeng-Eng Local time decay for a nonlinear beam equation. (English) Zbl 1210.35251 Methods Appl. Anal. 11, No. 1, 65-68 (2004). For the nonlinear beam equation \[ u_{tt}+ \Delta^2u+ f(u)= 0, \] where \(u= u(x,t)\), \(x=(x_1, x_2,\dots, x_n)\in\mathbb{R}^n\), \(\mathbb{R}^n\) is the \(n\)-dimensional Euclidean space \(n> 5\), \(t\geq 0\), \(\Delta=\) Laplacian in \(x\) and \(f(u)\) satisfies \[ c_1(uf(u)- 2F(u))+ c_0 u^2\geq F(u)\geq c_0 u^2,\;{dF\over du}= f(u), \] \(F(0)= 0\), \(c_0\), \(c_1\) are positive constants, it is shown that the local energy of solution \(u\) is integrable in time and the local \(L^2\) norm of solution \(u\) approaches zero as \(t\to\infty\). To prove this statement the Morawetz’s radial identity is used. Reviewer: István Ecsedi (Miskolc-Egyetemváros) Cited in 7 Documents MSC: 35Q74 PDEs in connection with mechanics of deformable solids 35R10 Partial functional-differential equations 35B40 Asymptotic behavior of solutions to PDEs 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74H40 Long-time behavior of solutions for dynamical problems in solid mechanics Keywords:nonlinear beam equation; local energy; time decay PDFBibTeX XMLCite \textit{J.-E. Lin}, Methods Appl. Anal. 11, No. 1, 65--68 (2004; Zbl 1210.35251) Full Text: DOI