Thermoelastic contact with Barber's heat exchange condition Kevin T. Andrews Peter Shi Meir Shillor Steve Wright Article

DOI :
10.1007/BF01188756

Cite this article as: Andrews, K.T., Shi, P., Shillor, M. et al. Appl Math Optim (1993) 28: 11. doi:10.1007/BF01188756
Abstract We consider a nonlinear parabolic problem that models the evolution of a one-dimensional 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 heat-exchange 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 heat-exchange coefficient as both a single-valued function and as a measurable selection from a maximal monotone graph. Both of these models represent modified versions of so-called 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.

Key words Thermoelastic contact Nonlinear heat-transfer coefficient Nonlinear boundary conditions Maximal monotone graph Signorini's condition

AMS classification Primary 35K60 Secondary 73C35 73T05 Communicated by D. Kinderlehrer

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Authors and Affiliations Kevin T. Andrews Peter Shi Meir Shillor Steve Wright 1. Department of Mathematical Sciences Oakland University Rochester USA