PORTUGALIAE MATHEMATICA Vol. 58, No. 2, pp. 233254 (2001) 

On A Class of Second Order Ode with A Typical Degenerate NonlinearityAlain Haraux and Qingxu YanUniversité P. et M. Curie, Analyse Numérique, Tour 5565, 5ème étage,4 pl. Jussieu, 75252 Paris cedex 05  FRANCE Yantai Teachers University, Department of Mathematics, Yantai, Shandong 264 025  P.R. CHINA Abstract: Global solutions of the second order ODE: $u''+u'+f(u)=0$ are studied where $f$ is a $C^1$ function satisfying $f(0)=0$, $f(u)>0$ for all $u\not=0$, $f(u)=o(\vert u\vert)$ as $u\rightarrow 0$; a typical case is $f(u)=c\,u^2$ or more generally $f(u)=c\,u^{\alpha}$ with $c>0$, $\alpha>1$. It is shown that all global solutions $u$ on $[0,+\infty)$ are bounded with $u'+u>0$ and $\lim_{t\rightarrow\infty}\{u(t)+u'(t)+u''(t)\}=0$. Moreover if $f(s)=c\,s^{\alpha}$ for some $c>0$, $\alpha>1$, there exists a unique global maximal negative solution $u_\in C^2(0,+\infty)$ and a unique global maximal solution $u_+\in C^2(0,+\infty)$ such that $\Sup_{t\in(0,+\infty)}u_+$ achieves its maximum value. The set of initial data giving rise to global trajectories for $t\geq 0$ is the unbounded closed domain ${\cal D}$ enclosed by the union of the two trajectories of $ u_+$ and $u_$ in the phase plane. Finally it is shown that $\meas({\cal D})<\infty$. Full text of the article:
Electronic version published on: 9 Feb 2006. This page was last modified: 27 Nov 2007.
© 2001 Sociedade Portuguesa de Matemática
