Tunisian Journal of Mathematics

On the ultimate energy bound of solutions to some forced second-order evolution equations with a general nonlinear damping operator

Alain Haraux

Abstract

Under suitable growth and coercivity conditions on the nonlinear damping operator $g$ which ensure nonresonance, we estimate the ultimate bound of the energy of the general solution to the equation $ü(t)+Au(t)+g(u̇(t))=h(t)$, $t∈ℝ+$, where $A$ is a positive selfadjoint operator on a Hilbert space $H$ and $h$ is a bounded forcing term with values in $H$. In general the bound is of the form $C(1+∥h∥4)$, where $∥h∥$ stands for the $L∞$ norm of $h$ with values in $H$ and the growth of $g$ does not seem to play any role. If $g$ behaves like a power for large values of the velocity, the ultimate bound has quadratic growth with respect to $∥h∥$ and this result is optimal. If $h$ is antiperiodic, we obtain a much lower growth bound and again the result is shown to be optimal even for scalar ODEs.

Article information

Source
Tunisian J. Math., Volume 1, Number 1 (2019), 59-72.

Dates
Revised: 30 October 2017
Accepted: 14 November 2017
First available in Project Euclid: 2 March 2019

https://projecteuclid.org/euclid.tunis/1551495681

Digital Object Identifier
doi:10.2140/tunis.2019.1.59

Mathematical Reviews number (MathSciNet)
MR3907734

Zentralblatt MATH identifier
07027517

Citation

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

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