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Thermoelastic contact with Barber's heat exchange condition

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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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Oakland University, 48309-4401, Rochester, MI, USA

    Kevin T. Andrews, Peter Shi, Meir Shillor & Steve Wright

Authors
  1. Kevin T. Andrews
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  2. Peter Shi
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  3. Meir Shillor
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  4. Steve Wright
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Communicated by D. Kinderlehrer

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Andrews, K.T., Shi, P., Shillor, M. et al. Thermoelastic contact with Barber's heat exchange condition. Appl Math Optim 28, 11–48 (1993). https://doi.org/10.1007/BF01188756

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