Solving the Linear 1D Thermoelasticity Equations with Pure Delay

TL;DR

This paper introduces a delayed PDE model for 1D thermoelasticity, proves well-posedness, shows convergence to classical solutions as delay vanishes, and provides explicit solution formulas.

Contribution

It presents a novel delayed PDE system for thermoelasticity, establishes its well-posedness, and connects it to classical models through convergence analysis.

Findings

Global well-posedness of the delayed system

Solution convergence to classical thermoelasticity as delay approaches zero

Explicit solution representation for the delay problem

Abstract

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.