On A Class of Second Order Ode with A Typical Degenerate Nonlinearity

Alain Haraux and Qingxu Yan

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 $\meas({\cal D})<\infty$.

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