Long-time asymptotic profiles of the n-D heat equation and modified heat kernels

TL;DR

This work develops a refined long-time description of the $n$-dimensional heat equation by introducing modified heat kernels $\hat{G}^{(k)}_t$ that incorporate the moments $\mathcal{M}_\alpha(f)$ of the initial data through spatial shifts $x^{*,\alpha}$ and time shifts $t^{*,\alpha}$. By performing a systematic Taylor expansion of the heat kernel and enforcing Condition A on higher-order moments, the authors show that the solution $u(x,t)$ can be approximated by $\hat{G}^{(k)}_t$ with explicit $L^p$-decay rates, and they extend the construction to degenerate cases via higher-order kernels. The key contributions include explicit error decompositions, rigorous $L^p$ decay estimates, and constructive examples of initial data that satisfy the required moment conditions, thereby linking initial data memory to long-time diffusion profiles. This framework generalizes prior modified-kernel approaches and provides precise quantitative asymptotics useful for applications in diffusion-dominated processes.

Abstract

We construct a long-time asymptotic profile to the initial value problem of the n-dimensional heat equation. Specifically, we present a modified heat kernel as a long-time asymptotic profile which changes the mass, the center of mass and the variance of the n-dimensional heat kernel in accordance with the moments of the initial data.

Theorems & Definitions (14)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Proposition 1.7
• Proposition 1.8
• Corollary 1.9
• Theorem 1.10