Applied Mathematics and Optimization

, Volume 28, Issue 1, pp 11–48 | Cite as

Thermoelastic contact with Barber's heat exchange condition

  • Kevin T. Andrews
  • Peter Shi
  • Meir Shillor
  • Steve Wright


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 


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Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • Kevin T. Andrews
    • 1
  • Peter Shi
    • 1
  • Meir Shillor
    • 1
  • Steve Wright
    • 1
  1. 1.Department of Mathematical SciencesOakland UniversityRochesterUSA

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