Schauder bases and the decay rate of the heat equation

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 $1\le p<\infty$ 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 $n\ge n_m$, 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.