Tunisian Journal of Mathematics
- Tunisian J. Math.
- Volume 1, Number 1 (2019), 59-72.
On the ultimate energy bound of solutions to some forced second-order evolution equations with a general nonlinear damping operator
Under suitable growth and coercivity conditions on the nonlinear damping operator which ensure nonresonance, we estimate the ultimate bound of the energy of the general solution to the equation , , where is a positive selfadjoint operator on a Hilbert space and is a bounded forcing term with values in . In general the bound is of the form , where stands for the norm of with values in and the growth of does not seem to play any role. If behaves like a power for large values of the velocity, the ultimate bound has quadratic growth with respect to and this result is optimal. If is antiperiodic, we obtain a much lower growth bound and again the result is shown to be optimal even for scalar ODEs.
Tunisian J. Math., Volume 1, Number 1 (2019), 59-72.
Received: 25 August 2017
Revised: 30 October 2017
Accepted: 14 November 2017
First available in Project Euclid: 2 March 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 34A34: Nonlinear equations and systems, general 34D20: Stability 35B40: Asymptotic behavior of solutions 35L10: Second-order hyperbolic equations 35L90: Abstract hyperbolic equations
Haraux, Alain. On the ultimate energy bound of solutions to some forced second-order evolution equations with a general nonlinear damping operator. Tunisian J. Math. 1 (2019), no. 1, 59--72. doi:10.2140/tunis.2019.1.59. https://projecteuclid.org/euclid.tunis/1551495681