Asymptotically free property of the solutions of an abstract linear hyperbolic equation with time-dependent coefficients

Abstract

This paper is concerned with an abstract dissipative hyperbolic equation with time-dependent coefficient. Under an assumption which ensures that the energy does not decay, this paper provides a condition on the coefficient, which is necessary and sufficient so that the solutions tend to the solutions of the free wave equation.

Keywords

abstract linear hyperbolic equation asymptotic behavior asymptotically free property

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References

[1]

Arosio , Asymptotic behavior as

t \to + \infty

of the solutions of linear hyperbolic equations with coefficients discontinuous in time (on a bounded domain), J. Differential Equations 39 (1981), 291–309.

[2]

Bárta , A generation theorem for hyperbolic equations with coefficients of bounded variation in time, Riv. Mat. Univ. Parma (7) 9 (2008), 17–30.

[3]

E.A.

Coddington and

Levinson , Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

[4]

Colombini ,

De Giorgi and

Spagnolo , Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps, Ann. Scuola Norm. Sup. Pisa 6 (1979), 511–559.

[5]

De Simon and

Torelli , Linear second order differential equations with discontinuous coefficients in Hilbert spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1 (1974), 131–154.

[6]

M.R.

Ebert ,

Fitriana and

Hirosawa , On the energy estimates of the wave equation with time dependent propagation speed asymptotically monotone functions, J. Math. Anal. Appl. 432 (2015), 654–677.

[7]

Hartman , Ordinary Differential Equations, 2nd edn, SIAM, Philadelphia, 2002.

[8]

Matsuyama , Asymptotic behavior for wave equation with time-dependent coefficients, Annali dell’Universita di Ferrara 52 (2006), 383–393.

[9]

Matsuyama , Asymptotic profiles for the Kirchhoff equation, Rend. Lincei Mat. Appl. 17 (2006), 377–395.

10.

[10]

Matsuyama and

Ruzhansky , Scattering for strictly hyperbolic systems with time-dependent coefficients, Math. Nachr. 286 (2013), 1191–1207.

11.

[11]

Ruzhansky and

Wirth , Dispersive estimates for hyperbolic systems with time-dependent coefficients, J. Differential Equations 251 (2011), 941–969.

12.

[12]

Ruzhansky and

Wirth, Asymptotic behaviour of solutions to hyperbolic partial differential equations, in: Variable Lebesgue Spaces and Hyperbolic Systems, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer, Basel, 2014, pp. 91–169.

13.

[13]

Wintner , Asymptotic integrations of adiabatic oscillator, Amer. J. Math. 69 (1947), 251–272.

14.

[14]

Yamazaki , Hyperbolic–parabolic singular perturbation for quasilinear equations of Kirchhoff type with weak dissipation, Math. Methods Appl. Sci. 32 (2009), 1893–1918.

15.

[15]

Yamazaki , Necessary condition on the dissipative term for the asymptotically free property of dissipative Kirchhoff equations, Nonlinear Analysis 125 (2015), 90–112.