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. We show that given a weighted L^p-space L_w^p({mathbb {R}}^N) with 1 le p < infty and a fast-growing weight w, there are Schauder bases (e_n)_{n=1}^infty in L_ w^p({mathbb {R}}^N) with the following property: given a positive integer m, there exists n_m > 0 such that, if the initial data f belong 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}. Such a basis can be constructed as a perturbation of any given Schauder basis. The proof is based on a construction of a basis of L_w^p( {mathbb {R}}^N), which annihilates an infinite sequence of bounded functionals.