The Very Fast Solution of a Special Second Order ODE with Exponentially Decaying Forcing and Applications

Haraux, Alain

Haraux, Alain

Bollettino dell'Unione Matematica Italiana, Tome 5 (2012), p. 233-241 / Harvested from Biblioteca Digitale Italiana di Matematica

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Résumé

Let $b$ , $c$ , $p$ be arbitrary positive constants and let $f\in C(\mathbb{R}^{+})$ be such that for some $\lambda>c$ , $F>0$ we have $|f(t)|\leq F\exp(-\lambda t)$ . Then all solutions $x$ of $x^{\prime\prime}+cx^{\prime}+b|x|^{p}x=f(t)$ tend to 0 as well as $x^{\prime}$ as $t$ tends to infinity. Moreover there exists a unique solution $y$ of (E) such that for some constant $C>0$ we have $|y(t)|+|y^{\prime}(t)|\leq C\exp(-\lambda t)$ for all $t>0$ . Finally all other solutions of (E) decay to 0 either like $e^{-ct}$ or like $(1+t)^{-1/p}$ as $t$ tends to infinity.

Publié le : 2012-06-01

MR 2977247 | Zbl 1260.34103

@article{BUMI_2012_9_5_2_233_0,
     author = {Alain Haraux},
     title = {The Very Fast Solution of a Special Second Order ODE with Exponentially Decaying Forcing and Applications},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {5},
     year = {2012},
     pages = {233-241},
     zbl = {1260.34103},
     mrnumber = {2977247},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2012_9_5_2_233_0}
}

Haraux, Alain. The Very Fast Solution of a Special Second Order ODE with Exponentially Decaying Forcing and Applications. Bollettino dell'Unione Matematica Italiana, Tome 5 (2012) pp. 233-241. http://gdmltest.u-ga.fr/item/BUMI_2012_9_5_2_233_0/

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