zbMATH — the first resource for mathematics

On a nonlinear coupled system of thermoelastic type with acoustic boundary conditions. (English) Zbl 1362.35046
Summary: 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.

35B40 Asymptotic behavior of solutions to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
74F05 Thermal effects in solid mechanics
Full Text: DOI
[1] Beale, JT; Rosencrans, SI, Acoustic boundary conditions, Bull Am Math Soc, 80, 1276-1278, (1974) · Zbl 0294.35045
[2] Beale, JT, Spectral properties of an acoustic boundary condition, Indiana Univ Math J, 25, 895-917, (1976) · Zbl 0325.35060
[3] Chen, G, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J Math Pures Appl, 58, 249-274, (1979) · Zbl 0414.35044
[4] Chipot, M; Lovat, B, On the asymptotic behaviour of some nonlocal problems, Positivity, 3, 65-81, (1999) · Zbl 0921.35071
[5] Clark, HR, Global existence, uniqueness and exponential stability for a nonlinear theermoelastic system, Appl Anal, 66, 39-56, (1997) · Zbl 0886.35023
[6] Clark, HR; Jutuca, LPSG, Miranda MM (1998) on a mixed problem for a linear coupled system with variable coefficients, Eletron J Differ Equ, 4, 1-20, (1998)
[7] Cousin, AT; Frota, CL; Larkin, NA, On a system of Klein-Gordon type equations with acoustic boundary conditions, J Math Anal Appl, 293, 293-309, (2004) · Zbl 1060.35118
[8] Dafermos, CM, On the existence and the asymptotic stability of solutons to the equations of linear thermoelasticity, Arch Ration Mech Anal, 29, 241-271, (1968) · Zbl 0183.37701
[9] Frigeri, S, Attractors for semilinear damped wave equations with an acoustic boundary condition, J Evol Equ, 10, 29-58, (2010) · Zbl 1239.35025
[10] Frota, CL; Goldstein, JA, Some nonlinear wave equations with acoustic boundary conditions, J Differ Equ, 164, 92-109, (2000) · Zbl 0979.35105
[11] Frota, CL; Larkin, NA, Uniform stabilization for hyperbolic equation with acoustic boundary conditions in simple connected domains, Prog Nonlinear Differ Equ Appl, 66, 297-312, (2005) · Zbl 1105.35018
[12] Frota, CL; Medeiros, LA; Vicente, A, Wave equation in domains with non-locally reacting boundary, Differ Integr Equ, 24, 1001-1020, (2011) · Zbl 1249.35221
[13] Graber, PJ, Uniform boundary stabilization of wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping, J Evol Equ, 12, 141-164, (2012) · Zbl 1250.35134
[14] Hansen, SW, Exponential energy decay in linear thermoelastic rod, J Math Anal Appl, 167, 429-442, (1992) · Zbl 0755.73012
[15] Haraux A, Zuazua E (1988) Decay estimates for some semilinear damped hyperbolic problems. Arch Ration Mech Anal 191-206 · Zbl 0654.35070
[16] Henry, D; Lopes, O; Perisinitto, A, Linear thermoelasticity: asymptotic stability and essential spectrum, Nonlinear analysis. TMA, 21, 65-75, (1993) · Zbl 0837.73010
[17] Kobayashi, Y; Tanaka, N, An application of semigroups of locally Lipschitz operators to carrier equations with acoustic boundary conditions, J Math Anal Appl, 338, 852-872, (2008) · Zbl 1145.47044
[18] Komornik, V; Zuazua, E, A direct method for boundary stabilization of the wave equation, J Math Pure Appl, 69, 33-54, (1990) · Zbl 0636.93064
[19] Limaco, J; Clark, HR; Frota, CL; Medeiros, LA, On an evolution equation with acoustic boundary conditions, Math Methods Appl Sci, 34, 2047-2059, (2011) · Zbl 1230.35058
[20] Lions JL (1969) Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires. Dunod, Paris · Zbl 0189.40603
[21] Medeiros, LA; Milla Miranda, M, On a boundary value problem for wave equations: existence, uniqueness-asymptotic behavior, Rev Mate Apl (Univerdidade de Chile), 17, 47-73, (1996) · Zbl 0859.35070
[22] Morse PM, Ingard KU (1968) Theoretical acoustic. McGraw-Hill, New York
[23] Mugnolo, D, Abstract wave equations with acoustic boundary conditions, Math Nachr, 279, 299-318, (2006) · Zbl 1109.47035
[24] Slemrod, M, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch Ration Mech Anal, 76, 97-133, (1981) · Zbl 0481.73009
[25] Schwartz L (1966) Thèorie des distributions, 3rd edn. Hermann, Paris · Zbl 0149.09501
[26] Vicente, A; Frota, CL, On a mixed problem with a nonlinear acoutic boundary condition for a non-locally reacting boundaries, J Math Anal Appl, 407, 328-338, (2013) · Zbl 1310.35153
[27] Vicente, A; Frota, CL, Nonlinear wave equation with weak dissipative term in domains with non-locally reacting boundary, Wave Motion, 50, 162-169, (2013) · Zbl 1360.76311
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.