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

TL;DR

This paper constructs a modified heat kernel as a long-time asymptotic profile for the n-dimensional heat equation, accounting for initial data moments to describe the solution's long-term behavior.

Contribution

It introduces a novel modified heat kernel that adjusts for initial data moments, providing a refined asymptotic description of solutions.

Findings

Modified heat kernel accurately captures long-time behavior

Adjusts for initial data moments like mass, center, and variance

Enhances understanding of asymptotic profiles in heat equations

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.