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

Haraux, Alain. "The Very Fast Solution of a Special Second Order ODE with Exponentially Decaying Forcing and Applications." Bollettino dell'Unione Matematica Italiana 5.2 (2012): 233-241. <http://eudml.org/doc/290968>.

@article{Haraux2012,
abstract = {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 \begin\{equation*\} \tag\{E\} x'' + cx' + b|x|^\{p\}x = f(t) \end\{equation*\} tend to 0 as well as $x'$ as $t$ tends to infinity. Moreover there exists a unique solution $y$ of (E) such that for some constant $C > 0$ we have $|y(t)| + |y'(t)| \leq C \exp(-\lambda t)$ 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.},
author = {Haraux, Alain},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {233-241},
publisher = {Unione Matematica Italiana},
title = {The Very Fast Solution of a Special Second Order ODE with Exponentially Decaying Forcing and Applications},
url = {http://eudml.org/doc/290968},
volume = {5},
year = {2012},
}

TY - JOUR
AU - Haraux, Alain
TI - The Very Fast Solution of a Special Second Order ODE with Exponentially Decaying Forcing and Applications
JO - Bollettino dell'Unione Matematica Italiana
DA - 2012/6//
PB - Unione Matematica Italiana
VL - 5
IS - 2
SP - 233
EP - 241
AB - 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 \begin{equation*} \tag{E} x'' + cx' + b|x|^{p}x = f(t) \end{equation*} tend to 0 as well as $x'$ as $t$ tends to infinity. Moreover there exists a unique solution $y$ of (E) such that for some constant $C > 0$ we have $|y(t)| + |y'(t)| \leq C \exp(-\lambda t)$ 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.
LA - eng
UR - http://eudml.org/doc/290968
ER -