Solving the Linear 1D Thermoelasticity Equations with Pure Delay

We propose a system of partial differential equations with a single constant delay \(\tau > 0\) describing the behavior of a one-dimensional thermoelastic solid occupying a bounded interval of \(\mathbb{R}^{1}\). For an initial-boundary value problem associated with this system, we prove a global well-posedness result in a certain topology under appropriate regularity conditions on the data. Further, we show the solution of our delayed model to converge to the solution of the classical equations of thermoelasticity as \(\tau \to 0\). Finally, we deduce an explicit solution representation for the delay problem.