We consider a class of nonlinear beam equations in the whole space $\mathbb R^n$. Using previous important work due to Levandovsky and Strauss we prove that, locally, the $H^1$-norm of a strong solution approaches zero as $t \to +\infty$ as long as the spatial dimension $n \ge 6$. The problem remains open for dimensions $1 \le n \le 5$.
"Time behavior for a class of nonlinear beam equations." Differential Integral Equations 19 (1) 15 - 29, 2006. https://doi.org/10.57262/die/1356050530