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On a nonlinear coupled system of thermoelastic type with acoustic boundary conditions

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Abstract

This paper deals with the existence, uniqueness, and asymptotic behavior of global solutions for a parabolic–hyperbolic coupled system with both local and nonlocal nonlinearities under mixed nonlinear acoustic boundary conditions.

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References

  • Beale JT, Rosencrans SI (1974) Acoustic boundary conditions. Bull Am Math Soc 80(6):1276–1278

    Article  MathSciNet  MATH  Google Scholar 

  • Beale JT (1976) Spectral properties of an acoustic boundary condition. Indiana Univ Math J 25:895–917

    Article  MathSciNet  MATH  Google Scholar 

  • Chen G (1979) Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain. J Math Pures Appl 58:249–274

    MathSciNet  MATH  Google Scholar 

  • Chipot M, Lovat B (1999) On the asymptotic behaviour of some nonlocal problems. Positivity 3(1):65–81

    Article  MathSciNet  MATH  Google Scholar 

  • Clark HR (1997) Global existence, uniqueness and exponential stability for a nonlinear theermoelastic system. Appl Anal 66:39–56

    Article  MathSciNet  Google Scholar 

  • Clark HR, Jutuca LPSG (1998) Miranda MM (1998) On a mixed problem for a linear coupled system with variable coefficients. Eletron J Differ Equ 4:1–20

    Google Scholar 

  • Cousin AT, Frota CL, Larkin NA (2004) On a system of klein-Gordon type equations with acoustic boundary conditions. J Math Anal Appl 293:293–309

    Article  MathSciNet  MATH  Google Scholar 

  • Dafermos CM (1968) On the existence and the asymptotic stability of solutons to the equations of linear thermoelasticity. Arch Ration Mech Anal 29:241–271

    Article  MATH  Google Scholar 

  • Frigeri S (2010) Attractors for semilinear damped wave equations with an acoustic boundary condition. J Evol Equ 10:29–58

    Article  MathSciNet  MATH  Google Scholar 

  • Frota CL, Goldstein JA (2000) Some nonlinear wave equations with acoustic boundary conditions. J Differ Equ 164:92–109

    Article  MathSciNet  MATH  Google Scholar 

  • Frota CL, Larkin NA (2005) Uniform stabilization for hyperbolic equation with acoustic boundary conditions in simple connected domains. Prog Nonlinear Differ Equ Appl 66:297–312

    MathSciNet  MATH  Google Scholar 

  • Frota CL, Medeiros LA, Vicente A (2011) Wave equation in domains with non-locally reacting boundary. Differ Integr Equ 24(11–12):1001–1020

    MathSciNet  MATH  Google Scholar 

  • Graber PJ (2012) Uniform boundary stabilization of wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping. J Evol Equ 12:141–164

    Article  MathSciNet  MATH  Google Scholar 

  • Hansen SW (1992) Exponential energy decay in linear thermoelastic rod. J Math Anal Appl 167:429–442

    Article  MathSciNet  MATH  Google Scholar 

  • Haraux A, Zuazua E (1988) Decay estimates for some semilinear damped hyperbolic problems. Arch Ration Mech Anal 191–206

  • Henry D, Lopes O, Perisinitto A (1993) Linear thermoelasticity: asymptotic stability and essential spectrum. Nonlinear analysis. TMA 21(1):65–75

    Article  Google Scholar 

  • Kobayashi Y, Tanaka N (2008) An application of semigroups of locally Lipschitz operators to carrier equations with acoustic boundary conditions. J Math Anal Appl 338:852–872

    Article  MathSciNet  MATH  Google Scholar 

  • Komornik V, Zuazua E (1990) A direct method for boundary stabilization of the wave equation. J Math Pure Appl 69:33–54

    MathSciNet  MATH  Google Scholar 

  • Limaco J, Clark HR, Frota CL, Medeiros LA (2011) On an evolution equation with acoustic boundary conditions. Math Methods Appl Sci 34:2047–2059

    Article  MathSciNet  MATH  Google Scholar 

  • Lions JL (1969) Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires. Dunod, Paris

    MATH  Google Scholar 

  • Medeiros LA, Milla Miranda M (1996) On a boundary value problem for wave equations: existence, uniqueness-asymptotic behavior. Rev Mate Apl (Univerdidade de Chile) 17:47–73

    MathSciNet  MATH  Google Scholar 

  • Morse PM, Ingard KU (1968) Theoretical acoustic. McGraw-Hill, New York

    Google Scholar 

  • Mugnolo D (2006) Abstract wave equations with acoustic boundary conditions. Math Nachr 279(3):299–318

    Article  MathSciNet  MATH  Google Scholar 

  • Slemrod M (1981) Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity. Arch Ration Mech Anal 76:97–133

    Article  MathSciNet  MATH  Google Scholar 

  • Schwartz L (1966) Thèorie des distributions, 3rd edn. Hermann, Paris

    MATH  Google Scholar 

  • Vicente A, Frota CL (2013) On a mixed problem with a nonlinear acoutic boundary condition for a non-locally reacting boundaries. J Math Anal Appl 407:328–338

    Article  MathSciNet  MATH  Google Scholar 

  • Vicente A, Frota CL (2013) Nonlinear wave equation with weak dissipative term in domains with non-locally reacting boundary. Wave Motion 50:162–169

    Article  MathSciNet  Google Scholar 

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Correspondence to H. R. Clark.

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Communicated by Geraldo Diniz.

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Braz e Silva, P., Clark, H.R. & Frota, C.L. On a nonlinear coupled system of thermoelastic type with acoustic boundary conditions. Comp. Appl. Math. 36, 397–414 (2017). https://doi.org/10.1007/s40314-015-0236-1

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  • DOI: https://doi.org/10.1007/s40314-015-0236-1

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