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

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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.

Article information

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

Dates
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
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

Subjects
Primary: 34A34: Nonlinear equations and systems, general 34D20: Stability 35B40: Asymptotic behavior of solutions 35L10: Second-order hyperbolic equations 35L90: Abstract hyperbolic equations

Keywords
second-order equation nonlinear damping energy bound antiperiodic

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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References

  • F. Aloui, I. Ben Hassen, and A. Haraux, “Compactness of trajectories to some nonlinear second order evolution equations and applications”, J. Math. Pures Appl. $(9)$ 100:3 (2013), 295–326.
  • L. Amerio and G. Prouse, “Uniqueness and almost-periodicity theorems for a non linear wave equation”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. $(8)$ 46 (1969), 1–8.
  • M. Biroli, “Bounded or almost periodic solution of the non linear vibrating membrane equation”, Ricerche Mat. 22 (1973), 190–202.
  • M. Biroli and A. Haraux, “Asymptotic behavior for an almost periodic, strongly dissipative wave equation”, J. Differential Equations 38:3 (1980), 422–440.
  • H. Brezis, “Équations et inéquations non linéaires dans les espaces vectoriels en dualité”, Ann. Inst. Fourier $($Grenoble$)$ 18:1 (1968), 115–175.
  • H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Notas de Matemática 50, North-Holland, Amsterdam, 1973.
  • A. Haraux, Nonlinear evolution equations: global behavior of solutions, Lecture Notes in Mathematics 841, Springer, 1981.
  • A. Haraux, “Dissipativity in the sense of Levinson for a class of second-order nonlinear evolution equations”, Nonlinear Anal. 6:11 (1982), 1207–1220.
  • A. Haraux, “Two remarks on hyperbolic dissipative problems”, pp. 161–179 in Nonlinear partial differential equations and their applications (Paris, 1983–1984), edited by H. Brezis and J.-L. Lions, Res. Notes in Math. 122, Pitman, Boston, 1985.
  • A. Haraux, Semi-linear hyperbolic problems in bounded domains, Math. Rep. $($Chur$)$ 3, part 1, Harwood, London, 1987.
  • A. Haraux, “Anti-periodic solutions of some nonlinear evolution equations”, Manuscripta Math. 63:4 (1989), 479–505.
  • A. Haraux, Systèmes dynamiques dissipatifs et applications, Recherches en Mathématiques Appliquées 17, Masson, Paris, 1991.
  • A. Haraux and E. Zuazua, “Decay estimates for some semilinear damped hyperbolic problems”, Arch. Rational Mech. Anal. 100:2 (1988), 191–206.
  • G. Prouse, “Soluzioni quasi-periodiche dell'equazione non omogenea delle onde, con termine dissipativo non lineare, I”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. $(8)$ 38 (1965), 804–807.
  • G. Prouse, “Soluzioni quasi-periodiche dell'equazione non omogenea delle onde, con termine dissipativo non lineare, II”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. $(8)$ 39 (1965), 11–18.
  • G. Prouse, “Soluzioni quasi-periodiche dell'equazione non omogenea delle onde, con termine dissipativo non lineare, III”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. $(8)$ 39 (1965), 155–160.
  • G. Prouse, “Soluzioni quasi-periodiche dell'equazione non omogenea delle onde, con termine dissipativo non lineare, IV”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. $(8)$ 39 (1965), 240–244.
  • E. Zuazua, “Stability and decay for a class of nonlinear hyperbolic problems”, Asymptotic Anal. 1:2 (1988), 161–185.