Sharp ultimate velocity bounds for the general solution of some linear second order evolution equation with damping and bounded forcing. (English) Zbl 07423283

In the paper under review, the authors prove the equivalence of different notions of optimal bound of solutions to a class of abstract evolution equations (Theorem 2.2). Furthermore, they investigate the optimal velocity bound for both scalar equations \[u^{\prime\prime}(t)+cu^\prime(t)+bu(t)=f(t)\] and vector equations \[u^{\prime\prime}(t)+cu^\prime(t)+Au(t)=f(t).\] The paper is well-written and the subject is of a significant interest.


34G10 Linear differential equations in abstract spaces
37C60 Nonautonomous smooth dynamical systems
34D05 Asymptotic properties of solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
Full Text: DOI arXiv


[1] Aloui, F.; Haraux, A., Sharp ultimate bounds of solutions to a class of second order linear evolution equations with bounded forcing term, J. Funct. Anal., 265, 10, 2204-2225 (2013) · Zbl 1287.34026
[2] Amerio, L.; Prouse, G., Uniqueness and almost-periodicity theorems for a non linear wave equation, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. (8), 46, 1-8 (1969) · Zbl 0176.40403
[3] Chu, J.; Garrione, M.; Gazzola, F., Stability analysis in some strongly prestressed rectangular plates, Evol. Equ. Control Theory, 9, 1, 275-299 (2020) · Zbl 1441.35038
[4] Fitouri, C.; Haraux, A., Sharp estimates of bounded solutions to some semilinear second order dissipative equations, J. Math. Pures Appl. (9), 92, 3, 313-321 (2009) · Zbl 1189.34111
[5] Fitouri, C.; Haraux, A., Boundedness and stability for the damped and forced single well Duffing equation, Discrete Contin. Dyn. Syst., 33, 1, 211-223 (2013) · Zbl 1269.34042
[6] Garrione, M.; Gazzola, F., Nonlinear Equations for Beams and Degenerate Plates with Piers, SpringerBriefs in Applied Sciences and Technology (2019), Springer: Springer Cham, ©2019. PoliMI SpringerBriefs · Zbl 1444.35006
[7] Gazzola, F., Mathematical models for suspension bridges, (Nonlinear Structural Instability. Nonlinear Structural Instability, MS&A. Modeling, Simulation and Applications, vol. 15 (2015), Springer: Springer Cham) · Zbl 1325.00032
[8] Giraudo, C., Optimal ultimate bound for linear second order dissipative equations (2019), University of Pisa, Bachelor thesis
[9] Haraux, A., Nonlinear Evolution Equations—Global Behavior of Solutions, Lecture Notes in Mathematics, vol. 841 (1981), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0461.35002
[10] Haraux, A., Uniform decay and Lagrange stability for linear contraction semi-groups, Mat. Apl. Comput., 7, 3, 143-154 (1988) · Zbl 0676.47020
[11] Haraux, A., On the double well Duffing equation with a small bounded forcing term, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 29, 207-230 (2005)
[12] (Kovacic, I.; Brennan, M. J., The Duffing Equation (2011), John Wiley & Sons, Ltd.: John Wiley & Sons, Ltd. Chichester), Nonlinear oscillators and their behaviour · Zbl 1220.34002
[13] Levitan, B. M.; Zhikov, V. V., Almost Periodic Functions and Differential Equations (1982), Cambridge University Press: Cambridge University Press Cambridge-New York, Translated from the Russian by L.W. Longdon · Zbl 0499.43005
[14] Loud, W. S., Boundedness and convergence of solutions of \(x'' + c x^\prime + g(x) = e(t)\), Duke Math. J., 24, 63-72 (1957) · Zbl 0077.09002
[15] Reed, M.; Simon, B., Methods of modern mathematical physics. I, (Functional Analysis (1980), Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers] New York)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.