Schauder bases and the decay rate of the heat equation

TL;DR

This paper constructs Schauder bases in weighted L^p spaces that enable control over the decay rates of solutions to the heat equation, revealing how initial data structure influences long-term behavior.

Contribution

It introduces a method to build Schauder bases in weighted L^p spaces that determine the decay rate of heat equation solutions based on initial data.

Findings

Existence of Schauder bases with prescribed decay properties for heat solutions

Decay rate of solutions can be at least t^{-m} for initial data in certain subspaces

Generalization of results to higher dimensions with weaker conditions

Abstract

We consider the classical Cauchy problem for the linear heat equation and integrable initial data in the Euclidean space $\mathbb{R}^N$. In the case $N=1$ we show that given a weighted $L^p$-space $L_w^p(\mathbb{R})$ with $1 \leq p < \infty$ and a fast growing weight $w$, there is a Schauder basis $(e_n)_{n=1}^\infty$ in $L_ w^p(\mathbb{R})$ with the following property: given a positive integer $m $ there exists $n_m > 0$ such that, if the initial data $f$ belongs to the closed linear space of $e_n$ with $n \geq n_m$, then the decay rate of the solution of the heat equation is at least $t^{-m}$. The result is also generalized to the case $N >1$ with a slightly weaker formulation. The proof is based on a construction of a Schauder basis of $L_w^p( \mathbb{R}^N)$, which annihilates an infinite sequence of bounded functionals.