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

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.