oaioai:CiteSeerX.psu:10.1.1.599.6778

Nova S¶erie ON A CLASS OF SECOND ORDER ODE WITH A TYPICAL DEGENERATE NONLINEARITY

Abstract

Abstract: Global solutions of the second order ODE: u00+u0+f(u) = 0 are studied where f is a C1 function satisfying f(0) = 0, f(u)> 0 for all u 6 = 0, f(u) = o(juj) as u! 0; a typical case is f(u) = c u2 or more generally f(u) = c juj ® with c> 0, ®> 1. It is shown that all global solutions u on [0;+1) are bounded with u0 + u> 0 and lim t!1 fju(t)j+ ju0(t)j+ ju00(t)jg = 0. Moreover if f(s) = c jsj ® for some c> 0, ®> 1, there exists a unique global maximal negative solution u ¡ 2 C2(0;+1) and a unique global maximal solution u+ 2 C2(0;+1) such that Supt2(0;+1) u+ achieves its maximum value. The set of initial data giving rise to global trajectories for t ¸ 0 is the unbounded closed domain D enclosed by the union of the two trajectories of u+ and u ¡ in the phase plane. Finally it is shown that meas(D) <1. 1 { Introduction and main results In this paper we study the second order ODE (1:1) u00 + u0 + f(u) = 0

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oaioai:CiteSeerX.psu:10.1.1.599.6778Last time updated on 10/29/2017

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