A note on the delayed heat equation: Instability with respect to initial data

Working in the context of a simple, one-dimensional, initial-boundary value problem involving homogeneous Dirichlet boundary data, we show that the time delayed heat equation can exhibit a type of instability with respect to the initial condition (IC); specifically, we show that a slight (in the L2L2 sense) change in the IC can change a well-posed problem to an ill-posed one. We also establish that a physically realistic solution is possible only if the IC is of a (very) specific form. The main implication of this study is that the single and dual phase lag models, which have been put forward as possible alternatives to Fourier’s law, are not valid constitutive relations for the thermal flux vector.