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Time behavior for a class of nonlinear beam equations. (English) Zbl 1212.35476

Summary: We consider a class of nonlinear beam equations in the whole space \(\mathbb R^n\). Using previous important work due to S. P. Levandosky and W. A. Strauss [Methods Appl. Anal. 7, No. 3, 479–487 (2000; Zbl 1029.35182)] we prove that, locally, the \(H^1\)-norm of a strong solution approaches zero as \(t\to +\infty \) as long as the spatial dimension \(n>6\). The problem remains open for dimensions \(1\leq n\leq 5\).

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
35B40 Asymptotic behavior of solutions to PDEs
35D35 Strong solutions to PDEs
35L75 Higher-order nonlinear hyperbolic equations
74H40 Long-time behavior of solutions for dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)

Citations:

Zbl 1029.35182
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