Htykuut

Consider the nonlinear ODE
$$tag{1}
frac{d^2u}{dt^2}+u=u^3, qquad tinmathbb R,$$
which has the conserved quantity
$$tag{2}
E=frac12 u'^2+frac12 u^2 -frac14u^4.$$
Consider only these solutions that satisfy $u>0$ and $u'>0$.

I am interested in solutions that blow-up at time $T$, in the sense that
$$
lim_{t nearrow T} u(t)=+infty.$$
The following family describes all solutions with $E=0$, parametrized by the blow-up time;
$$
u_{0, T}(t):=frac{sqrt{2}}{sin(T-t)}.$$
The phase portrait suggests that the trajectories of these solutions are an attractor for (1);

and this is reasonable, because (2) yields
$$
u'=sqrt{2E -u^2 + frac12u^4} = frac{u^2}{sqrt2} + O(1), $$
so when $u$ is very big I expect $u$ to be indistinguishable from the unique solution to
$v'=frac{v^2}{sqrt 2} $ that blows up at $T$, that is, $$v_T(t)=frac{sqrt2}{T-t}, $$
and $v_T$ is asymptotically equivalent to $u_{0, T}$ as $tnearrow T$.

I would like to obtain a more precise, quantitative version of this attraction.

Question. Let $u$ be a solution to (1) such that $u(t)to +infty$ as $tnearrow T$. Is it true that $$|u(t)-u_{0, T}(t)|to 0,qquad text{as }tnearrow T?$$ Is it true that $$frac{u(t)}{u_{0,T}(t)}to 1,qquad text{as }tnearrow T?$$

Remark. The equation (1) is a special case of Duffing equation.