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Local time decay for a nonlinear beam equation. (English) Zbl 1210.35251

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.

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
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