Revisit on How to Derive Asymptotic Profiles to Some Evolution Equations

TL;DR

This paper presents a simple method to derive asymptotic profiles of solutions to heat and damped wave equations with weighted initial data, enhancing understanding of their long-term behavior.

Contribution

It introduces a straightforward approach for deriving asymptotic profiles for evolution equations, improving upon existing methods.

Findings

Effective asymptotic profiles derived for heat and damped wave equations

Applicable to solutions with weighted L^{1,1} initial data

Simplifies previous derivation techniques

Abstract

We consider the Cauchy problem in ${\bf R}^{n}$ for heat and damped wave equations. We derive asymptotic profiles to those solutions with weighted $L^{1,1}({\bf R}^{n})$ data by presenting a simple method.