(Related to this Math.SE question.)

For $p>1$, let $u$ be a solution to $$\tag{1}\frac{d^2 u}{dt^2} + u = |u|^{p-1}u$$ that blows up at $T>0$, that is $$\lim_{t\nearrow T}u(t)=+\infty.$$

*Remark*. If $u$ solves (1), then the following quantity is independent on $t$; $$\tag{2} E=\frac12\left(\frac{du}{dt}\right)^2 +\frac{u^2}{2} -\frac{|u|^{p+1}}{p+1}.$$

Question. Is it true that $$\lim_{t\nearrow T} \frac{u(t)}{C(T-t)^{-\alpha}}=1,\qquad \text{where }C=\left(2\frac{p+1}{(p-1)^2}\right)^\frac1{p-1}\text{ and }\ \alpha=\frac{2}{p-1}? $$

I believe that the answer is affirmative. My motivation is the heuristic that $u_0:=C(T-t)^{-\alpha}$ should solve both $$\tag{3}\frac{d^2 u_0}{dt^2} =|u_0|^{p-1}u_0, $$ and $$\tag{4}\left(\frac{du_0}{dt}\right)^2 = \frac{2}{p+1}|u_0|^{p+1}, $$ and that is indeed the case.

The equations (3) and (4) are obtained from (1) and (2) by considering the highest power of $u$ only.