Vol. 1, No. 1, 2019

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

Alain Haraux

Vol. 1 (2019), No. 1, 59–72
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 + , 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 + h4), 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.

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Keywords
second-order equation, nonlinear damping, energy bound, antiperiodic
Mathematical Subject Classification 2010
Primary: 34A34, 34D20, 35B40, 35L10, 35L90
Milestones
Received: 25 August 2017
Revised: 30 October 2017
Accepted: 14 November 2017
Published: 26 January 2018
Authors
Alain Haraux
Sorbonne Universités, UPMC (Paris VI) & CNRS
Laboratoire Jacques-Louis Lions
Paris
France