## Differential and Integral Equations

### Critical exponents, special large-time behavior and oscillatory blow-up in nonlinear ODE's

Philippe Souplet

#### Abstract

We provide a complete study of the large-time behavior of all solutions of the equation $u''+|u|^{p-1}u=|u'|^{q-1}u'$, $t\geq 0$, for all values of the exponents $p,\, q>1$. We find the existence of two critical curves ${q=p}$, and ${q=2p/(p+1)}$, separating three regions of the plane ${(p,q)}$ for which the global behaviors exhibit a sharp contrast.

(i) If ${q\geq p}$, all nontrivial solutions blow up in finite time at the rate ${u'(t)\approx (T-t)^{-1/(q-1)}}$, except for an exceptional (unbounded) global solution, unique up to a sign and a time-translation.

(ii) If ${2p/(p+1)<q<p}$, all nontrivial solutions blow up in finite time. They do so at the same rate as above, except for an exceptional solution, unique up to a sign and a time-translation, which blowsup at the (faster) rate ${u'(t)\approx (T-t)^{-p/(p-q)}}$.

(iii) If ${1<q \leq 2p/(p+1)}$, all nontrivial solutions have oscillatory finite time blow-up, and the energy blows up at the rate ${E(t)\approx (T-t)^{-2/(q-1)}}$.

As a generalization of the case ${q\geq p}$, we prove that the second-order ODE ${u''=F(u,u')}$, ${t\geq 0}$ admits global solutions for all values of ${u(0)}$, whenever ${F}$, loosely speaking, does not grow faster with respect to ${u}$ than with respect to ${u'}$, with ${F}$ having the sign of ${u'}$ for ${u'}$ large. Among the techniques used in this paper are the method of variable time-translations, introduced in a previous work of the author, phase-plane analysis, Liapunov functions, and suitable shooting methods.

#### Article information

Source
Differential Integral Equations, Volume 11, Number 1 (1998), 147-167.

Dates
First available in Project Euclid: 1 May 2013