On some dynamic thermal non clamped contact problems

We study a class of dynamic thermal sub-differential contact problems with friction, for long memory visco-elastic materials, without the clamped condition, which can be put into a general model of system defined by a second order evolution inequality, coupled with a first order evolution equation. We present and establish an existence and uniqueness result, by using general results on first order evolution inequality, with monotone operators and fixed point methods. Finally a fully discrete scheme for numerical approximations is provided, and corresponding various numerical computations in dimension two will be given.