1. The universal bound property for a class of second order ODEs
- Author
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Mama Abdelli, Alain Haraux, Université Djilali Liabès [Sidi-Bel-Abbès], Laboratoire Jacques-Louis Lions (LJLL), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
General Mathematics ,AMS classification numbers: 34A34 ,010102 general mathematics ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,Ode ,Dynamical Systems (math.DS) ,34D05 ,[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] ,01 natural sciences ,34C10 ,Decay rate ,010101 applied mathematics ,Crystallography ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Mathematics - Dynamical Systems ,0101 mathematics ,34E99 Keywords: Second order scalar ODE ,Mathematics - Abstract
We consider the scalar second order ODE u + |u | $\alpha$ u + |u| $\beta$ u = 0, where $\alpha$, $\beta$ 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 < $\alpha$ < $\beta$ and moreover there is a constant C independent of the initial data such that $\forall$t > 0, u (t) 2 + |u(t)| $\beta$+2 $\le$ C max{t -- 2 $\alpha$ , t -- ($\alpha$+1)($\beta$+2) $\beta$--$\alpha$ }.
- Published
- 2018