American Institute of Mathematical Sciences

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2012, 2(1): 91-104. doi: 10.3934/naco.2012.2.91

Solvability of a class of thermal dynamical contact problems with subdifferential conditions

Samir Adly 1, , Oanh Chau 2, and  Mohamed Rochdi 2,

1. 

Institut d'Ingénierie Informatique de Limoges and LACO, URA-1586, 123 Avenue A. Thomas, 87060 Limoges Cedex
2. 

Université de La Réunion, Département Maths-Info, BP 7151, 15 Avenue René Cassin, 97715 Saint Denis Messag, cedex 09, La Réunion

Received  January 2011 Revised  August 2011 Published  March 2012

We study a class of dynamic thermal sub-differential contact problems with friction, for long memory visco-elastic materials, which can be put into a general model of system defined by a second order evolution inequality, coupled with a first order evolution equation. We present and establish an existence and uniqueness result, by using general results on first order evolution inequality, with monotone operators and fixed point methods.
Keywords: sub-differential contact condition, Long memory thermo-visco-elasticity, dynamical process, evolution inequality., fixed point.
Mathematics Subject Classification: Primary: 74M10, 74M15, 74F25, 74H20, 74H25, 34G2.
Citation: Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91
References:
[1]

B. Awbi and O. Chau, Quasistatic Thermovisoelastic Frictional Contact Problem with Damped Response,, Applicable Analysis, 83 (2004), 635. doi: 10.1080/00036810410001657233. Google Scholar

[2]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Editura Academiei, (1976). Google Scholar

[3]

H. Brézis, Problèmes unilatéraux,, J. Math. Pures et Appli., 51 (1972), 1. Google Scholar

[4]

O. Chau and M. Rochdi, On a dynamic bilateral contact problem with friction for viscoelastic materials,, Int. J. of Appli. Math. and Mech., 2 (2006), 41. Google Scholar

[5]

P. G. Ciarlet, "Mathematical Elasticity, Vol. I, Three-Dimensional Elasticity,", North-Holland, (1988). Google Scholar

[6]

M. Cocou and G. Scarella, Existence of a solution to a dynamic unilateral contact problem for a cracked viscoelastic body,, C.R. Acad. Sci. Paris, 338 (2004), 341. Google Scholar

[7]

G. Duvaut and J. L. Lions, "Les Inéquations en Mécanique et en Physique,", Dunod, (1972). Google Scholar

[8]

Ch. Eck, J. Jarusek and M. Krbec, "Unilateral Contact Problems, Variational Methods and Existence Theorems,", Chapman and Hall, (2005). doi: 10.1201/9781420027365. Google Scholar

[9]

D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, "Variational and Hemivariational Inequalities, Theory, Methods and Applications, Volume I: Unilateral Analysis and Unilateral Mechanics,", Kluwer Academic Publishers, (2003). Google Scholar

[10]

J. Jarušek and Ch. Eck, Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions,, Mathematical Models and Methods in Applied Sciences, 9 (1999), 11. doi: 10.1142/S0218202599000038. Google Scholar

[11]

N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity,", SIAM, (1988). doi: 10.1137/1.9781611970845. Google Scholar

[12]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,", Dunod et Gauthier-Villars, (1969). Google Scholar

[13]

J. Necas, J. Jarusek and J. Haslinger, On the solution of variational inequality to the Signorini problem with small friction,, Bolletino U.M.I., 17 (1980), 796. Google Scholar

[14]

J. Nečas and I. Hlaváček, "Mathematical Theory of Elastic and Elastoplastic Bodies: An introduction,", Elsevier, (1981). Google Scholar

[15]

P. D. Panagiotopoulos, "Inequality Problems in Mechanics and Applications,", Birkhäuser, (1985). doi: 10.1007/978-1-4612-5152-1. Google Scholar

[16]

P. D. Panagiotopoulos, "Hemivariational Inequalities, Applications in Mechanics and Engineering,", Springer-Verlag, (1993). Google Scholar

[17]

Eberhard Zeidler, "Nonlinear Functional Analysis and its Applications, II/A, Linear Monotone Operators,", Springer Verlag, (1997). Google Scholar

show all references

References:
[1]

B. Awbi and O. Chau, Quasistatic Thermovisoelastic Frictional Contact Problem with Damped Response,, Applicable Analysis, 83 (2004), 635. doi: 10.1080/00036810410001657233. Google Scholar

[2]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Editura Academiei, (1976). Google Scholar

[3]

H. Brézis, Problèmes unilatéraux,, J. Math. Pures et Appli., 51 (1972), 1. Google Scholar

[4]

O. Chau and M. Rochdi, On a dynamic bilateral contact problem with friction for viscoelastic materials,, Int. J. of Appli. Math. and Mech., 2 (2006), 41. Google Scholar

[5]

P. G. Ciarlet, "Mathematical Elasticity, Vol. I, Three-Dimensional Elasticity,", North-Holland, (1988). Google Scholar

[6]

M. Cocou and G. Scarella, Existence of a solution to a dynamic unilateral contact problem for a cracked viscoelastic body,, C.R. Acad. Sci. Paris, 338 (2004), 341. Google Scholar

[7]

G. Duvaut and J. L. Lions, "Les Inéquations en Mécanique et en Physique,", Dunod, (1972). Google Scholar

[8]

Ch. Eck, J. Jarusek and M. Krbec, "Unilateral Contact Problems, Variational Methods and Existence Theorems,", Chapman and Hall, (2005). doi: 10.1201/9781420027365. Google Scholar

[9]

D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, "Variational and Hemivariational Inequalities, Theory, Methods and Applications, Volume I: Unilateral Analysis and Unilateral Mechanics,", Kluwer Academic Publishers, (2003). Google Scholar

[10]

J. Jarušek and Ch. Eck, Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions,, Mathematical Models and Methods in Applied Sciences, 9 (1999), 11. doi: 10.1142/S0218202599000038. Google Scholar

[11]

N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity,", SIAM, (1988). doi: 10.1137/1.9781611970845. Google Scholar

[12]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,", Dunod et Gauthier-Villars, (1969). Google Scholar

[13]

J. Necas, J. Jarusek and J. Haslinger, On the solution of variational inequality to the Signorini problem with small friction,, Bolletino U.M.I., 17 (1980), 796. Google Scholar

[14]

J. Nečas and I. Hlaváček, "Mathematical Theory of Elastic and Elastoplastic Bodies: An introduction,", Elsevier, (1981). Google Scholar

[15]

P. D. Panagiotopoulos, "Inequality Problems in Mechanics and Applications,", Birkhäuser, (1985). doi: 10.1007/978-1-4612-5152-1. Google Scholar

[16]

P. D. Panagiotopoulos, "Hemivariational Inequalities, Applications in Mechanics and Engineering,", Springer-Verlag, (1993). Google Scholar

[17]

Eberhard Zeidler, "Nonlinear Functional Analysis and its Applications, II/A, Linear Monotone Operators,", Springer Verlag, (1997). Google Scholar

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