Why does $u$ belong to $L^1$?

Question

I'm reading an article of second order gradient like-system $$u^{\prime\prime}+Bu^\prime+\nabla F(u) = 0. \tag{1.1}$$

(here $U_t$ is similar to $u^\prime$ in the above equation)

I don't understand why the author could conclude "$||U_t||^2\in L^1$". I think we can only say that $U_t(t)$ has a finit limit as $t$ tends to infinity.

Thank you very much.