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 studies of a quasi-static thermoelastic
problem involving a critical parameter $\alpha$.
For $\alpha <1$, 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 \rr \infty$ is examined,
show that the non-local term has no impact on the asymptotic behavior
for $ \alpha <1$. The paper concludes with some
remarks on the case $\alpha >1$.