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

Alain Haraux

Bollettino dell'Unione Matematica Italiana (2012)

  • Volume: 5, Issue: 2, page 233-241
  • ISSN: 0392-4041

Abstract

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Let b , c , p be arbitrary positive constants and let f C ( + ) be such that for some λ > c , F > 0 we have | f ( t ) | F exp ( - λ t ) . Then all solutions x of x ′′ + c x + b | x | p x = f ( t ) 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 ) | C exp ( - λ t ) for all t > 0 . Finally all other solutions of (E) decay to 0 either like e - c t or like ( 1 + t ) - 1 / p as t tends to infinity.

How to cite

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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 -

References

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  3. BEN HASSEN, I. - CHERGUI, L., Convergence of global and bounded solutions of some nonautonomous second order evolution equations with nonlinear dissipation, J. Dynam. Differential Equations, 23, no. 2 (2011), 315-332. Zbl1228.35148MR2802889DOI10.1007/s10884-011-9212-7
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  6. HARAUX, A., On the fast solution of evolution equations with a rapidly decaying source term, Math. Control & Rel. Fields1, (March 2011) 1-20. Zbl1227.34063MR2822682DOI10.3934/mcrf.2011.1.1
  7. HARAUX, A., Sharp decay estimates of the solutions to a class of nonlinear second order ODE, Analysis and applications (Singap.), 9, no. 1 (2011), 49-69. Zbl1227.34052MR2763360DOI10.1142/S021953051100173X
  8. HARAUX, A. - JENDOUBI, M. A., On a second order dissipative ODE in Hilbert space with an integrable source term, to appear in Acta Mathematica Scientia. Zbl1265.34186MR2921869DOI10.1016/S0252-9602(12)60009-5
  9. HUANG, S. Z. - TAKAC, P., Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal., 46, no. 5 (2001), Ser. A: Theory Methods, 675-698. Zbl1002.35022MR1857152DOI10.1016/S0362-546X(00)00145-0
  10. JENDOUBI, M. A. - MAY, R., On an asymptotically autonomous system with Tikhonov type regularizing term, Arch. Math., 95 (2010), 389-399. Zbl1217.34085MR2727316DOI10.1007/s00013-010-0181-6

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