In this paper the initial value problem and global properties of solutions are studied for the scalar second order ODE: (|u ′ | l u ′ ) ′ + c|u ′ | α u ′ + d|u| β u = 0, where α, β, l, c, d are positive constants. In particular, existence, uniqueness and regularity as well as optimal decay rates of solutions to 0 are obtained depending on the various parameters, and the oscillatory or non-oscillatory behavior is elucidated .
We consider the scalar second order ODE u ′′ + |u ′ | α u ′ + |u| β u = 0, where α, β are two positive numbers and the non-linear semi-group S(t) generated on IR 2 by the system in (u, u ′ ). We prove that S(t)IR 2 is bounded for all t > 0 whenever 0 < α < β and moreover there is a constant C independent of the initial data such that
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