A parabolic integro-differential equation arising from thermoelastic contact

In this paper we consider a class of integro-differential equations of parabolic type arising in the study of a quasi-static thermoelastic contact problem involving a critical parameter α. For α < 1, the problem is first transformed into an equivalent standard parabolic equation with non-local and non-linear boundary conditions. Then the existence, uniqueness and continuous dependence of the solution upon the data are demonstrated via solution representation techniques and the maximum principle. Finally the asymptotic behavior of the solution as t → ∞ is examined, and we show that the non-local term has no impact on the asymptotic behavior for α < 1. The paper concludes with some remarks on the case α > 1.