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On a nonlinear thermoelastic system with nonlocal coefficients. (English) Zbl 1327.35071
Summary: This paper deals with the global existence and uniqueness of solutions, and uniform stabilization of the energy of an initial-boundary value problem for a thermoelastic system with nonlinear terms of nonlocal kind. The asymptotic stabilization of the energy of system is obtained without any dissipation mechanism acting in the displacement variable \(u\) of the membrane equation.

35G61 Initial-boundary value problems for systems of nonlinear higher-order PDEs
74F05 Thermal effects in solid mechanics
35B40 Asymptotic behavior of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
Full Text: DOI
[1] Aubin, J. P., Un théorème de compacité, C. R. Math. Acad. Sci. Paris, 256, 5042-5044, (1963) · Zbl 0195.13002
[2] Brito, E. H., The damped elastic string equations generalized: existence, uniqueness, regularity and stability, Appl. Anal., 13, 219-233, (1982)
[3] Calvacanti, M. M.; Cavalcanti, V. N.D.; Martinez, P., Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203, 119-158, (2004) · Zbl 1049.35047
[4] Chipot, M.; Lovat, B., On the asymptotic behaviour of some nonlocal problems, Positivity, 3, 1, 65-81, (1999) · Zbl 0921.35071
[5] Clark, H. R., Global existence, uniqueness and exponential stability for a nonlinear thermoelastic system, Appl. Anal., 66, 39-56, (1997)
[6] Clark, H. R., Asymptotic and smoothness properties of a nonlinear equation with damping, Commun. Appl. Anal., 4, 321-337, (2000) · Zbl 1084.34537
[7] Clark, H. R.; San Gil Jutuca, L. P.; Miranda, M. M., On a mixed problem for a linear coupled system with variable coefficients, Electron. J. Differential Equations, 1998, 04, 1-20, (1998) · Zbl 0886.35043
[8] Cousin, A. T.; Frota, C. L.; Lar’kin, N. A., Global solvability and decay of the energy for nonhomogeneous Kirchhoff equation, Differential Integral Equations, 15, 10, 219-236, (2002) · Zbl 1011.35097
[9] Dafermos, C. M., On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Ration. Mech. Anal., 29, 241-271, (1968) · Zbl 0183.37701
[10] Fernandez-Cara, E.; Límaco, J.; Menezes, S. B., Null controllability for a parabolic equation with nonlocal nonlinearities, Systems Control Lett., 61, 107-111, (2012) · Zbl 1250.93031
[11] Goldstein, J. A., Semigroups and second order differential equations, J. Funct. Anal., 4, 1, 50-70, (1969) · Zbl 0179.14605
[12] Hansen, S. W., Exponential energy decay in linear thermoelastic rod, J. Math. Anal. Appl., 167, 429-442, (1992) · Zbl 0755.73012
[13] Henry, D.; Lopes, O.; Perisinitto, A., Linear thermoelasticity: asymptotic stability and essential spectrum, Nonlinear Anal., 21, 1, 65-75, (1993)
[14] Komornik, V.; Zuazua, E., A direct method for boundary stabilization of the wave equation, J. Math. Pures Appl., 69, 33-54, (1990) · Zbl 0636.93064
[15] Lasiecka, I.; Tataru, D., Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential Integral Equations, 6, 507-533, (1993) · Zbl 0803.35088
[16] Límaco, J.; Clark, H. R.; Medeiros, L. A., On damped Kirchhoff equation with variable coefficients, J. Math. Anal. Appl., 307, 641-655, (2005) · Zbl 1128.35371
[17] Límaco, J.; Clark, H. R.; Medeiros, L. A., Vibrations of elastic string with nonhomogeneous material, J. Math. Anal. Appl., 344, 806-820, (2008) · Zbl 1145.35435
[18] Lions, J. L., Quelques Méthodes de Résolutions des problèmes aux limites non-linéaires, (1969), Dunod Paris · Zbl 0189.40603
[19] Louredo, A. T.; Marinho, A. O.; Clark, M. R., Boundary stabilization for a coupled system, Nonlinear Anal., 74, 6988-7004, (2011) · Zbl 1230.35015
[20] Marinho, A. O.; Clark, M. R.; Clark, H. R., Existence and boundary stabilization of solutions for the coupled semilinear system, Nonlinear Anal., 70, 4226-4244, (2009) · Zbl 1172.35360
[21] Medeiros, L. A.; Límaco, J.; Menezes, S. B., Vibrations of elastic strings: mathematical aspects. part one, J. Comput. Anal. Appl., 4, 3, 91-127, (2002) · Zbl 1118.35335
[22] Medeiros, L. A.; Milla Miranda, M., On a boundary value problem for wave equations: existence, uniqueness-asymptotic behavior, Rev. Mat. Apl. Univ. Chile, 17, 47-73, (1996) · Zbl 0859.35070
[23] Milla Miranda, M.; Medeiros, L. A., On a nonlinear wave equation with damping, Rev. Mat. Complut., 3, 213-231, (1990) · Zbl 0721.35044
[24] Milla Miranda, M.; San Gil Jutuca, L. P., Existence and boundary stabilization of solutions for the Kirchhoff equation, Comm. Partial Differential Equations, 24, 1759-1880, (1999) · Zbl 0930.35110
[25] Quinn, J. P.; Russell, D. L., Asymptotic stability and energy decay rates for solutions of hyperbolic equation with boundary damping, Proc. Roy. Soc. Edinburgh Sect. A, 77, 97-127, (1977) · Zbl 0357.35006
[26] Schwartz, L., Thèorie des distributions, (1966), Hermann Paris
[27] Slemrod, M., Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch. Rot. Mech. Anal., 76, 97-133, (1981) · Zbl 0481.73009
[28] Vitillaro, E., Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186, 259-298, (2002)
[29] Zuazua, E., Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim., 28, 466-478, (1990) · Zbl 0695.93090
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