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.
Similar content being viewed by others
References
Beale JT, Rosencrans SI (1974) Acoustic boundary conditions. Bull Am Math Soc 80(6):1276–1278
Beale JT (1976) Spectral properties of an acoustic boundary condition. Indiana Univ Math J 25:895–917
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
Chipot M, Lovat B (1999) On the asymptotic behaviour of some nonlocal problems. Positivity 3(1):65–81
Clark HR (1997) Global existence, uniqueness and exponential stability for a nonlinear theermoelastic system. Appl Anal 66:39–56
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
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
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
Frigeri S (2010) Attractors for semilinear damped wave equations with an acoustic boundary condition. J Evol Equ 10:29–58
Frota CL, Goldstein JA (2000) Some nonlinear wave equations with acoustic boundary conditions. J Differ Equ 164:92–109
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
Frota CL, Medeiros LA, Vicente A (2011) Wave equation in domains with non-locally reacting boundary. Differ Integr Equ 24(11–12):1001–1020
Graber PJ (2012) Uniform boundary stabilization of wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping. J Evol Equ 12:141–164
Hansen SW (1992) Exponential energy decay in linear thermoelastic rod. J Math Anal Appl 167:429–442
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
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
Komornik V, Zuazua E (1990) A direct method for boundary stabilization of the wave equation. J Math Pure Appl 69:33–54
Limaco J, Clark HR, Frota CL, Medeiros LA (2011) On an evolution equation with acoustic boundary conditions. Math Methods Appl Sci 34:2047–2059
Lions JL (1969) Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires. Dunod, Paris
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
Morse PM, Ingard KU (1968) Theoretical acoustic. McGraw-Hill, New York
Mugnolo D (2006) Abstract wave equations with acoustic boundary conditions. Math Nachr 279(3):299–318
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
Schwartz L (1966) Thèorie des distributions, 3rd edn. Hermann, Paris
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
Vicente A, Frota CL (2013) Nonlinear wave equation with weak dissipative term in domains with non-locally reacting boundary. Wave Motion 50:162–169
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Geraldo Diniz.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-015-0236-1