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Journal of Dynamics and Differential Equations

, Volume 19, Issue 4, pp 915–933 | Cite as

Sharp Estimates of Bounded Solutions to Some Second-order Forced Dissipative Equations

  • Alain Haraux
Article

Keywords

second-order equation bounded solution 

Résumé

A l’aide d’inégalités différentielles, on établit une estimation proche de l’optimalité pour la norme dans \(L^\infty({\mathbb{R}},V)\) de l’unique solution bornée de u′′ + cu′ +  Auf(t) lorsque AA * ≥  λ I est un opérateur borné ou non sur un espace de Hilbert réel H, VD(A 1/2) et λ, c sont des constantes positives, tandis que \(f\in L^\infty({\mathbb{R}}, H) \) .

By using differential inequalities, a close-to-optimal \(L^\infty({\mathbb{R}},V)\) bound of the unique bounded solution of u′′ + cu′ +  Auf(t) is obtained whenever AA * ≥  λ I is a bounded or unbounded linear operator on a real Hilbert space H, VD(A 1/2) and λ, c are positive constants, while \(f\in L^\infty({\mathbb{R}}, H) \) .

AMS Classification Numbers

34C15 34C25 34C27 34D05 34D030 

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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsCNRS and Université Pierre et Marie CurieParis Cedex 05France

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