# American Institute of Mathematical Sciences

April  2011, 4(2): 273-309. doi: 10.3934/dcdss.2011.4.273

## Long-time behaviour of a thermomechanical model for adhesive contact

 1 Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia 2 Dipartimento di Matematica Università di Brescia, Via Valotti 9, I–25133 Brescia, Italy 3 Dipartimento di Matematica, Università di Brescia, Via Valotti 9, I–25133 Brescia

Received  April 2009 Published  November 2010

This paper deals with the large-time analysis of a PDE system modelling contact with adhesion, in the case when thermal effects are taken into account. The phenomenon of adhesive contact is described in terms of phase transitions for a surface damage model proposed by M. Frémond. Thermal effects are governed by entropy balance laws. The resulting system is highly nonlinear, mainly due to the presence of internal constraints on the physical variables and the coupling of equations written in a domain and on a contact surface. We prove existence of solutions on the whole time interval $(0,+\infty)$ by a double approximation procedure. Hence, we are able to show that solution trajectories admit cluster points which fulfil the stationary problem associated with the evolutionary system, and that in the large-time limit dissipation vanishes.
Citation: Elena Bonetti, Giovanna Bonfanti, Riccarda Rossi. Long-time behaviour of a thermomechanical model for adhesive contact. Discrete and Continuous Dynamical Systems - S, 2011, 4 (2) : 273-309. doi: 10.3934/dcdss.2011.4.273
##### References:
 [1] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 1976. [2] E. Bonetti, G. Bonfanti and R. Rossi, Global existence for a contact problem with adhesion, Math. Meth. Appl. Sci., 31 (2008), 1029-1064. doi: 10.1002/mma.957. [3] E. Bonetti, G. Bonfanti and R. Rossi, Well-posedness and long-time behaviour for a model of contact with adhesion, Indiana Univ. Math. J., 56 (2007), 2787-2819. doi: 10.1512/iumj.2007.56.3079. [4] E. Bonetti, G. Bonfanti and R. Rossi, Thermal effects in adhesive contact: Modelling and analysis, Nonlinearity, 22 (2009), 2697-2731. doi: 10.1088/0951-7715/22/11/007. [5] E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Global solution to a singular integrodifferential system related to the entropy balance, Nonlinear Anal., 66 (2007), 1949-1979. doi: 10.1016/j.na.2006.02.035. [6] E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1001-1026. doi: 10.3934/dcdsb.2006.6.1001. [7] E. Bonetti, P. Colli and M. Frémond, A phase field model with thermal memory governed by the entropy balance, Math. Models Methods Appl. Sci., 13 (2003), 1565-1588. doi: 10.1142/S0218202503003033. [8] E. Bonetti, M. Frémond and E. Rocca, A new dual approach for a class of phase transitions with memory: Existence and long-time behaviour of solutions, J. Math. Pures Appl., 88 (2007), 455-481. doi: 10.1016/j.matpur.2007.09.005. [9] H. Brézis, "Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert," North-Holland, Amsterdam, 1973. [10] P. Colli, On some doubly nonlinear evolution equations in Banach spaces, Japan J. Indust. Appl. Math., 9 (1992), 181-203. doi: 10.1007/BF03167565. [11] P. Colli, G. Gilardi, P. Laurençot and A. Novick-Cohen, Existence and long-time behavior of the conserved phase-field system with memory, Discrete Contin. Dynamic Systems, 5 (1999), 375-390. [12] E. Feireisl and G. Schimperna, Large time behaviour of solutions to Penrose-Fife phase change models, Math. Methods Appl. Sci., 28 (2005), 2117-2132. doi: 10.1002/mma.659. [13] M. Frémond, "Non-smooth Thermomechanics," Springer-Verlag, Berlin, 2002. [14] M. Grasselli, H. Petzeltová and G. Schimperna, Long-time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72. doi: 10.4171/ZAA/1277. [15] A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Masson, Paris, 1991. [16] P. Krejčí and S. Zheng, Pointwise asymptotic convergence of solutions for a phase separation model, Discrete Contin. Dyn. Syst., 16 (2006), 1-18. doi: 10.3934/dcds.2006.16.1. [17] P. Rybka and K-H. Hoffmann, Convergence of solutions to Cahn-Hilliard equation, Comm. Partial Differential equations, 24 (1999), 1055-1077. [18] J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.

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##### References:
 [1] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 1976. [2] E. Bonetti, G. Bonfanti and R. Rossi, Global existence for a contact problem with adhesion, Math. Meth. Appl. Sci., 31 (2008), 1029-1064. doi: 10.1002/mma.957. [3] E. Bonetti, G. Bonfanti and R. Rossi, Well-posedness and long-time behaviour for a model of contact with adhesion, Indiana Univ. Math. J., 56 (2007), 2787-2819. doi: 10.1512/iumj.2007.56.3079. [4] E. Bonetti, G. Bonfanti and R. Rossi, Thermal effects in adhesive contact: Modelling and analysis, Nonlinearity, 22 (2009), 2697-2731. doi: 10.1088/0951-7715/22/11/007. [5] E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Global solution to a singular integrodifferential system related to the entropy balance, Nonlinear Anal., 66 (2007), 1949-1979. doi: 10.1016/j.na.2006.02.035. [6] E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1001-1026. doi: 10.3934/dcdsb.2006.6.1001. [7] E. Bonetti, P. Colli and M. Frémond, A phase field model with thermal memory governed by the entropy balance, Math. Models Methods Appl. Sci., 13 (2003), 1565-1588. doi: 10.1142/S0218202503003033. [8] E. Bonetti, M. Frémond and E. Rocca, A new dual approach for a class of phase transitions with memory: Existence and long-time behaviour of solutions, J. Math. Pures Appl., 88 (2007), 455-481. doi: 10.1016/j.matpur.2007.09.005. [9] H. Brézis, "Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert," North-Holland, Amsterdam, 1973. [10] P. Colli, On some doubly nonlinear evolution equations in Banach spaces, Japan J. Indust. Appl. Math., 9 (1992), 181-203. doi: 10.1007/BF03167565. [11] P. Colli, G. Gilardi, P. Laurençot and A. Novick-Cohen, Existence and long-time behavior of the conserved phase-field system with memory, Discrete Contin. Dynamic Systems, 5 (1999), 375-390. [12] E. Feireisl and G. Schimperna, Large time behaviour of solutions to Penrose-Fife phase change models, Math. Methods Appl. Sci., 28 (2005), 2117-2132. doi: 10.1002/mma.659. [13] M. Frémond, "Non-smooth Thermomechanics," Springer-Verlag, Berlin, 2002. [14] M. Grasselli, H. Petzeltová and G. Schimperna, Long-time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72. doi: 10.4171/ZAA/1277. [15] A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Masson, Paris, 1991. [16] P. Krejčí and S. Zheng, Pointwise asymptotic convergence of solutions for a phase separation model, Discrete Contin. Dyn. Syst., 16 (2006), 1-18. doi: 10.3934/dcds.2006.16.1. [17] P. Rybka and K-H. Hoffmann, Convergence of solutions to Cahn-Hilliard equation, Comm. Partial Differential equations, 24 (1999), 1055-1077. [18] J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.
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