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 \(1\leq n\leq 5\).


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.)


Zbl 1029.35182