Critical exponents, special large-time behavior and oscillatory blow-up in nonlinear ODE’s. (English) Zbl 1015.34038

The author gives a complete classification of the long-time behavior in the interval \(t\geq 0\) of nontrivial solutions to autonomous second-order ODEs of the form \(u'' + |u|^{p-1}u = |u'|^{q-1}u'\) with exponents \(p, q > 1\). He finds three subdomains of the parameter domain separated by the curves \(q = p\) and \(q = 2p/(p+1)\) with qualitatively different properties. A global solution exists (and is unique up to the sign and a time translation) if and only if \(q\geq p\); all the other solutions blow up in a finite time. The blow-up is oscillatory if and only if \(1<q\leq 2p/(p+1)\), and the blow-up rates are different in each subdomain.


34D05 Asymptotic properties of solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations