Prove that $ \lim_{t\to +\infty} ||u^\prime(t)|| = 0$.

Question

I'm really confused how can we conclude $\displaystyle \lim_{t\to +\infty} ||u^\prime(t)|| = 0$ in the following paragraph:

I tried to prove when $N = 1$, but things are also difficult to handle. The author also wrote in his article that in case $N=1$, all solutions of equation $(1.1)$ are monotonic, but I do not know how to prove it either.