Strong and Mild Extrapolated $L^2$-Solutions to the Heat Equation with Constant Delay

TL;DR

This paper develops a Hilbert space framework for solving the heat equation with delay, establishing well-posedness, explicit solutions, and energy decay, while highlighting issues with lower order regularizations.

Contribution

It introduces a novel solution theory for delayed heat equations with boundary conditions, including explicit solution representation and decay analysis.

Findings

Well-posedness under weak regularity assumptions

Explicit solution representation provided

Exponential energy decay in dissipative cases

Abstract

We propose a Hilbert space solution theory for a nonhomogeneous heat equation with delay in the highest order derivatives with nonhomogeneous Dirichlet boundary conditions in a bounded domain. Under rather weak regularity assumptions on the data, we prove a well-posedness result and give an explicit representation of solutions. Further, we prove an exponential decay rate for the energy in the dissipative case. We also show that lower order regularizations lead to ill-posedness, also for higher-order equations. Finally, an application with physically relevant constants is given.