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Lions Regularization of the Telegraph System with Nonlinear Boundary Conditions

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Abstract

The purpose of this paper is to study a Lions type regularization of the telegraph system with nonlinear boundary conditions. An asymptotic expansion of order zero for the solution of this regularization is established, including some boundary layer corrections. Specifically, under some appropriate smoothness and compatibility conditions on the data an estimate for the remainder term of the expansion is derived with respect to the \(C([0,T];L^2(0,1)^2)\) norm. Our main theorem generalizes a result regarding the particular case of homogeneous linear boundary conditions reported by Apreutesei and Djafari Rouhani (Nonlinear Anal 35:3049–3061, 2010).

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References

  1. Aftabizadeh, A.R., Pavel, N.H.: Boundary value problems for second order differential equations and a convex problem of Bolza. Differ. Integral Equ. 2, 495–509 (1989)

    MathSciNet  MATH  Google Scholar 

  2. Ahsan, M., Moroşanu, G.: Elliptic-like regularization of semilinear evolution equations. J. Math. Anal. Appl. 396, 759–771 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ahsan, M., Moroşanu, G.: Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations. J. Differ. Equ. 257, 2926–2949 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Apreutesei, N.C., Djafari Rouhani, B.: Elliptic regularization for the semilinear telegraph system. Nonlinear Anal. 35, 3049–3061 (2010)

    Article  MATH  Google Scholar 

  5. Barbu, L., Moroşanu, G.: Elliptic-like regularization of semilinear evolution equations and applications to some hyperbolic problems. J. Math. Anal. Appl. (to appear)

  6. Barbu, L., Moroşanu, G.: Elliptic-like regularization of a fully nonlinear evolution inclusion and applications. Commun. Contemp. Math. doi:10.1142/S0219199716500371

  7. Barbu, L., Moroşanu, G.: Singularly Perturbed Boundary Value Problems. Birkhäuser, Boston (2007)

    MATH  Google Scholar 

  8. Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leyden (1976)

    Book  MATH  Google Scholar 

  9. Barbu, V.: Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic Press, New York (1993)

    MATH  Google Scholar 

  10. Barbu, V.: Nonlinear boundary value problems for a class of hyperbolic systems. Rev. Roumaine Math. Pures Appl. 22(2), 155–168 (1977)

    MathSciNet  MATH  Google Scholar 

  11. Brezis, H.: Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Math. Studies 5. North Holland, Amsterdam (1973)

  12. Cooke, K.L., Krumme, D.W.: Differential-difference equations and nonlinear initial boundary value problems for linear hyperbolic partial differential equations. J. Math. Anal. Appl. 24, 372–387 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lions, J.L.: Perturbations singulières dans les problèmes aux limites et en contrôl optimal, Lecture Notes in Math. 323. Springer, Berlin (1973)

    Book  Google Scholar 

  14. Hokkanen, V.-M., Moroşanu, G.: Functional Methods in Differential Equations. Chapman & Hall/CRC, London (2002)

    Book  MATH  Google Scholar 

  15. Khatibzadeh, H., Moroşanu, G.: Strong and weak solutions to second order differential inclusions governed by monotone operators. Set-Valued Var. Anal. 22(2), 521–531 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, W.-H.: Differential Equations of Hydraulics, Transients, Dispersion, and Groundwater Flow. Prentice-Hall, Upper Saddle River (1972)

    Google Scholar 

  17. Marinov, C.A., Neittaanmäki, P.: Mathematical Models in Electrical Circuits: Theory and Applications. Kluwer, Dordrecht (1991)

    Book  MATH  Google Scholar 

  18. Moroşanu, G.: Nonlinear Evolution Equations and Applications. D. Reidel, Dordrecht (1988)

    MATH  Google Scholar 

  19. Showalter, R.E.: Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations. Amer. Math. Soc., Providence (1997)

    MATH  Google Scholar 

  20. Streeter, L.V., Wyle, E.B.: Hydraulic Transients. McGraw-Hill, New York (1967)

    Google Scholar 

  21. Vasilieva, A.B., Butuzov, V.F., Kalashev, I.V.: The Boundary Function Method for Singular Perturbation Problems. SIAM, Philadelphia (1995)

    Book  Google Scholar 

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  1. Faculty of Mathematics and Computer Science, Ovidius University, 124 Mamaia Blvd, 900527, Constanţa, Romania

    Luminiţa Barbu

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  1. Luminiţa Barbu
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Correspondence to Luminiţa Barbu.

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Barbu, L. Lions Regularization of the Telegraph System with Nonlinear Boundary Conditions. Results Math 72, 731–745 (2017). https://doi.org/10.1007/s00025-017-0670-z

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  • DOI: https://doi.org/10.1007/s00025-017-0670-z

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