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Periodic propagation of singularities for heat equations with time delay

Gengsheng Wang, Huaiqiang Yu, Yubiao Zhang

TL;DR

This work analyzes the heat equation with time delay $\partial_t y = \Delta y + a\,y(t-\tau,x)$ on a bounded domain with Dirichlet boundary, revealing a novel $\tau$-periodic, bidirectional propagation of singularities along the time axis, with a stepwise increase of the joint-derivative order by 2 per period in forward time and a decrease by 2 in backward time. Central to the analysis is an explicit semigroup-based representation of the delayed flow, which separates the classical heat dynamics from the delay-driven memory term and enables precise transfer of local regularity properties. The authors establish necessary and sufficient compatibility conditions for the initial data and history that govern the periodic propagation, and they provide numerical simulations that illustrate the predicted recurrence of singularities at lattice times $\{j\tau\}$. The results highlight a hyperbolic-like behavior induced by time delay in an otherwise parabolic model and clarify how initial data and history jointly shape the singularity dynamics. These insights deepen understanding of singularity propagation in delay PDEs and suggest broader implications for controllability and observability in delayed diffusion systems.

Abstract

This paper presents two remarkable phenomena associated with the heat equation with a time delay: namely, the propagation of singularities and periodicity. These are manifested through a distinctive mode of propagation of singularities in the solutions. Precisely, the singularities of the solutions propagate periodically in a bidirectional fashion along the time axis. Furthermore, this propagation occurs in a stepwise manner. More specifically, when propagating in the positive time direction, the order of the joint derivatives of the solution increases by 2 for each period; conversely, when propagating in the reverse time direction, the order of the joint derivatives decreases by 2 per period. Additionally, we elucidate the way in which the initial data and historical values impact such a propagation of singularities. The phenomena we have discerned not only corroborate the pronounced differences between heat equations with and without time delay but also vividly illustrate the substantial divergence between the heat equation with a time delay and the wave equation, especially when viewed from the point of view of singularity propagation.

Periodic propagation of singularities for heat equations with time delay

TL;DR

This work analyzes the heat equation with time delay on a bounded domain with Dirichlet boundary, revealing a novel -periodic, bidirectional propagation of singularities along the time axis, with a stepwise increase of the joint-derivative order by 2 per period in forward time and a decrease by 2 in backward time. Central to the analysis is an explicit semigroup-based representation of the delayed flow, which separates the classical heat dynamics from the delay-driven memory term and enables precise transfer of local regularity properties. The authors establish necessary and sufficient compatibility conditions for the initial data and history that govern the periodic propagation, and they provide numerical simulations that illustrate the predicted recurrence of singularities at lattice times . The results highlight a hyperbolic-like behavior induced by time delay in an otherwise parabolic model and clarify how initial data and history jointly shape the singularity dynamics. These insights deepen understanding of singularity propagation in delay PDEs and suggest broader implications for controllability and observability in delayed diffusion systems.

Abstract

This paper presents two remarkable phenomena associated with the heat equation with a time delay: namely, the propagation of singularities and periodicity. These are manifested through a distinctive mode of propagation of singularities in the solutions. Precisely, the singularities of the solutions propagate periodically in a bidirectional fashion along the time axis. Furthermore, this propagation occurs in a stepwise manner. More specifically, when propagating in the positive time direction, the order of the joint derivatives of the solution increases by 2 for each period; conversely, when propagating in the reverse time direction, the order of the joint derivatives decreases by 2 per period. Additionally, we elucidate the way in which the initial data and historical values impact such a propagation of singularities. The phenomena we have discerned not only corroborate the pronounced differences between heat equations with and without time delay but also vividly illustrate the substantial divergence between the heat equation with a time delay and the wave equation, especially when viewed from the point of view of singularity propagation.

Paper Structure

This paper contains 18 sections, 12 theorems, 170 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Equation under study
  3. History, motivation and aim
  4. Notation, subspaces and concepts
  5. Plan of this paper
  6. Main results
  7. Local regularity properties for the heat equation
  8. Explicit expression of solutions
  9. Proofs of main results
  10. Proof of Theorem \ref{['20230510-yb-theorem-PropagationOfSingularitiesOnlyWithInitialdata']}
  11. Proof of Theorem \ref{['20230510-yb-theorem-PropagationOfSingularitiesForGeneralSolutions']}
  12. Proof of Theorem \ref{['20240406-yubiao-Theorem-SingularitiesOfHistoricalValueAtEndpoints']}
  13. Numerical simulations
  14. Appendices
  15. Proof of Lemma \ref{['20230524-yb-lemma-PropagationOfSmoothnessForHeatSolutions']}
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $y_0\in L^2(\Omega)$ and $\phi \in C_0^{\infty}((-\tau,0) \times \Omega)$. Then, for any $x_0\in \Omega$, $j\in \mathbb N$ and $s,\alpha\in \mathbb N$ with $\alpha \leq j$, the following three statements are equivalent: Furthermore, the solution $y(\cdot,\cdot;y_0,\phi)$ is smooth in the set $( (-\tau,+\infty) \setminus \{j\tau\}_{j=0}^{+\infty} ) \times \Omega$.

Figures (1)

  • Figure 1: Images of $\partial_t^{j_t} y(t+0,\cdot; \delta_{0.3}, 0)$ (drawn in blue) at six time instants.

Theorems & Definitions (34)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Corollary 3.3
  • Remark 3.4
  • ...and 24 more