  In the present paper, we study a model of a thermoelastic string that is
initially heated. We classify all the possible asymptotic states when time
tends to infinity of such a model. Actually, we show that whatever the initial
data is, a heated string must converge to a flat, steady string with uniformly
distributed heat. The latter distribution is calculated from the energy
conservation. In order to obtain the result, we need to take a few steps. In
the first two steps, time-independent bounds from above and from below (by a
positive constant) of the temperature are obtained. This is done via the
Moser-like iteration. The lower bound is obtained via the Moser iteration on
the negative part of the logarithm of temperature. In the third step, we obtain
a time-independent higher-order estimate, which yields compactness of a
sequence of the values of the solution when time tends to infinity. Here, an
estimate involving the Fisher information of temperature, together with a
recent functional inequality from \cite{CFHS} and an $L^2(L^2)$ estimate of the
gradient of entropy, enable us to arrive at a tricky Gr\"{o}nwall type
inequality. Finally, in the last steps, we define the dynamical system on a
proper functional phase space and study its $\omega$-limit set. To this end, we
use, in particular, the quantitative version of the second principle of
thermodynamics. Also, the entropy dissipation term and the bound of the entropy
from below are useful when identifying the structure of the $\omega$-limit set.
