The Thermoelastic and Viscoelastic Contact of Two Rods

We consider the thermoelastic and viscoelastic contact problem of two rods and prove the existence of a weak solution using a penalization method and compensated compactness. Moreover, for the thermoelastic contact we show that the weak solution converges to zero exponentially as time goes to infinity, and for the viscoelastic contact we prove that the weak solution decays to zero with the same rates as the relaxation functions do.