# The blow-up rate of a nonlinear oscillator

(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.

Your heuristics is correct. Indeed, suppose that $$u(t)\to\infty$$ as $$t\uparrow T$$. Then for some real $$h>0$$ and all $$t\in[T-h,T)$$ we have $$u(t)>0$$ and, by the "conservation of energy" stated in your post,
$$$$u'(t)^2=2E-u(t)^2+\frac{2u(t)^{p+1}}{p+1}\sim \frac{2u(t)^{p+1}}{p+1},$$$$ because $$p>1$$ and hence $$p+1>2$$; everywhere here, the asymptotic relations are given for $$t\uparrow T$$. Because $$u''$$ exists and $$u(t)\to\infty$$, we see that $$$$u'(t)\sim \sqrt{\frac{2}{p+1}} \,u(t)^{(p+1)/2},\quad -\frac d{dt}\,\big(u(t)^{-(p-1)/2}\big)\sim c_p:=\sqrt{\frac{2}{p+1}}\,\frac{p-1}2,$$$$ $$$$u(t)^{-(p-1)/2}=\int_t^T \Big(-\frac d{ds}\,\big(u(s)^{-(p-1)/2}\big)\Big)\, ds \sim c_p(T-t),$$$$ and thus $$$$u(t)\sim [c_p(T-t)]^{-2/(p-1)}=C(T-t)^{-\alpha}$$$$ for $$C=\big(2\frac{p+1}{(p-1)^2}\big)^\frac1{p-1}$$ and $$\alpha=\frac{2}{p-1}$$, as desired.