Thermoelastic contact with Barber's heat exchange condition
 Kevin T. Andrews,
 Peter Shi,
 Meir Shillor,
 Steve Wright
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We consider a nonlinear parabolic problem that models the evolution of a onedimensional thermoelastic system that may come into contact with a rigid obstacle. The mathematical problem is reduced to solving a nonlocal heat equation with a nonlinear and nonlocal boundary condition. This boundary condition contains a heatexchange coefficient that depends on the pressure when there is contact with the obstacle and on the size of the gap when there is no contact. We model the heatexchange coefficient as both a singlevalued function and as a measurable selection from a maximal monotone graph. Both of these models represent modified versions of socalled imperfect contact conditions found in the work of Barber. We show that strong solutions exist when the coefficient is taken to be a continuously differentiable function and that weak solutions exist when the coefficient is taken to be a measurable selection from a maximal monotone graph. The proofs of these results reveal an interesting interplay between the regularity of the initial condition and the behavior of the coefficient at infinity.
 Amann H (1978) Invariant sets and existence theorems for semilinear parabolic and elliptic systems. J Math Anal Appl 65:432–465
 Amann H (1984) Existence and regularity for semilinear parabolic equations. Ann Scuola Norm Sup Pisa XI:593–676
 Amann H (1988) Parabolic evolution equations and nonlinear boundary conditions. J Differential Equations 72:201–269
 Andrews KT, Mikelić A, Shi P, Shillor M, Wright S (1992) Onedimensional thermoelastic contact with a stressdependent radiation condition. SIAM J Math Anal 23:1393–1416
 Barber JR (1978) Contract problems involving a cooled punch. J Elasticity 8:409–423
 Barber JR (1987) Stability of thermoelastic contact. Proc IMechE International Conference on Tribology, London, pp 981–986
 Barber JR (1981) Stability of thermoelastic contact for the Aldo model. Trans AMSE 48:555–558
 Barber JR, Dundurs J, Comninou M (1980) Stability considerations in thermoelastic contact. J Appl Mech 47:871–874
 Barber JR, Zhang R (1988) Transient behaviour and stability for the thermoelastic contact of two rods of dissimilar materials. Internat J Mech Sci 30(9):691–704
 Benilan PH, Crandall MG, Sacks (1988) Some L^{1} existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions. Appl Math Optim 17:203–224
 Brezis H (1977) Operateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hubert. NorthHolland, Amsterdam
 Cannon JR (1984) The One Dimensional Heat Equation. Encyclopedia of Mathematics and Its Applications. AddisonWesley, Menlo Park
 Cannon Jr, Lin Y, de Heok V (1990) A quasilinear parabolic equation with nonlocal boundary condition. Rend Mat 10:239–264
 Carlson DE (1972) Linear thermoelasticity. In: Flugge S (ed), Handbuch der Physik, vol VIa/2. SpringerVerlag, Berlin, pp 297–345
 Comninou M, Dundurs J (1979) On the Barber boundary conditions for thermoelastic contact. J Appl Mech 46:849–853
 Dafermos CM (1968) On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity. Arch Rational Mech Anal 29:241–271
 Day WA (1985) Heat Conduction Within Linear Thermoelasticity. SpringerVerlag, New York
 Day WA (1988) Justification of the uncoupled and quasistatic approximations in a problem of dynamic thermoelasticity. Arch Rational Mech Anal 80:135–158
 Diaz JI (1985) Nonlinear Partial Differential Equations and Free Boundaries. Research Notes in Mathematics, vol 106. Pitman, Boston
 Diaz JI, Jimenez RF (1988) Boundary behavior of solutions of the Signorini problem, I: The elliptic case. Boll Un Mat Ital B 7(2):127–139
 van Duijn CJ, Hulshof J (1987) An ellipticparabolic problem with a nonlocal boundary condition. ]Arch Rational Mech Anal 99(1):61–73
 Duvaut G (1980) Free boundary problems connected with thermoelasticity and unilateral contact. Free Boundary Problems: Proceedings of a Seminar held in Pavia, September–October 1979, vol II. Instituto nazionale, Roma
 Duvaut G, Lions JL (1972) Inequations en thermoelasticite et magnetohydrodynamique. Arch Rational Mech Anal 46:241–279
 Duvaut G, Lions JL (1976) Inequalities in Mechanics and Physics. SpringerVerlag, Berlin
 Fichera G (1972) Boundary value problems of elasticity with unilateral constraints. In: Flugge S (ed), Handbuch der Physik, vol VIa/2. SpringerVerlag, Berlin, pp 391–424
 Gilbert RP, Shi P, Shillor M (1990) A quasistatic contact problem in linear thermoelasticity. Rend Mat (7) 10:785–808
 Hrusa V, Messaoudi SA (1990) On formation of singularities in one dimensional nonlinear thermoelasticity. Arch Rational Mech Anal 111(2):135–151
 Johnson KL (1985) Contact Mechanics. Cambridge University Press, Cambridge
 Kikuchi N, Oden JT (1988) Contact Problems in Elasticity. SIAM, Philadelphia, PA
 Kufner OJ, John J, Fucik S (1977) Function Spaces. Noordhoff, Leyden
 Ladyzenskaja OJ, Solonnikov VA, Uralceva NN (1968) Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence, RI
 Lions JL, Magenes E (1972) Nonhomogeneous Boundary Value Problems and Applications, II. SpringerVerlag, New York
 Nickell RE, Sackman JL (1968) Variational principles for linear coupled thermoelasticity. Quart Appl Math 26:11–26
 Primicerio M (private communication)
 Richmond O, Huang NC (1977) Interface stability during unidirectional solidification of a pure metal. Proc. 6th Canadian Congress of Applied Mechanics, pp 453–454
 Rodrigues JF (1987) Obstacle Problems in Mathematical Physics. NorthHolland, Amsterdam
 Rodrigues JF (1988) A unilateral thermolastic Stefantype problem. Portugal Math 45:91–103
 Rodrigues JF (preprint) Remarks on the Reynolds problem of elastohydrodynamic lubrication
 Rogers RC, Antman SS (1986) Steadystate problems of nonlinear electromagnetothermodynamics. Arch Rational Mech Anal 95(4):279–323
 Roubiček T (preprint) A coupled contact problem in nonlinear thermoviscoelasticity
 Shi P, Shillor M (1990) Uniqueness and stability of the solution to a thermoelastic contact problem. European J Appl Math 1:371–387
 Shi P, Shillor M (to appear) A quasistatic contact problem in thermoelasticity with a radiation condition for the temperature. J. Math Anal Appl
 Shi P, Shillor M (1992) Existence of a solution to thendimensional problem of thermoelastic contact. Comm Partial Differential Equations 17:1597–1618
 Shi P, Shillor M, Zou XL (1991) Numerical solutions to onedimensional problems of thermoelastic contact. Comput Math Appl 22(10):65–78
 Slemrod M (1981) Global existence, uniqueness, and asymptotic stability of classical smooth solutions in onedimensional nonlinear thermoelasticity. Arch Rational Mech Anal 76(2):97–133
 Srinivasan MG, France DM (1985) Nonuniqueness in steady — state heat transfer in prestressed duplex tubes — analysis and case history. J Appl mech 48:555–558
 Zhang R, Barber JR (1990) Effect of material properties on the stability of static thermoelastic contact. J Appl mech 57(2):365–369
 Title
 Thermoelastic contact with Barber's heat exchange condition
 Journal

Applied Mathematics and Optimization
Volume 28, Issue 1 , pp 1148
 Cover Date
 19930701
 DOI
 10.1007/BF01188756
 Print ISSN
 00954616
 Online ISSN
 14320606
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Thermoelastic contact
 Nonlinear heattransfer coefficient
 Nonlinear boundary conditions
 Maximal monotone graph
 Signorini's condition
 Primary 35K60
 Secondary 73C35
 73T05
 Authors

 Kevin T. Andrews ^{(1)}
 Peter Shi ^{(1)}
 Meir Shillor ^{(1)}
 Steve Wright ^{(1)}
 Author Affiliations

 1. Department of Mathematical Sciences, Oakland University, 483094401, Rochester, MI, USA