Seminar on Banach and Metric Space Geometry

Date: October 26, 2018

Time: 3:00PM - 4:00PM

Location: BLOC 220

Speaker: Jari Taskinen, University of Helsinki

Title: Schauder bases and the decay rate of the heat equation

Abstract: Joint work with José Bonet (Valencia) and Wolfgang Lusky (Paderborn) We consider the classical Cauchy problem for the linear heat equation with integrable initial data f = f(x) in the Euclidean space R^N \ni x. As well known, the conventional solution formula implies that for generic f, the sup-norm (w.r.t. x) of the solution u(x,t) decays at the speed rate t^{-N/2} for large times t. Faster decay rates are possible for special initial data. Our aim is to present a new approach to this phenomenon and also to show that initial data leading to faster decay rates is in a certain sense not so rare. Accordingly, given a weight function w = w(x) growing rapidly at the infinity, we construct Schauder bases (e_n)_{n=1}^\infty in the Banach space L_w^p (R^N) , 1 < p < \infty or p=1, with the following property: given an arbitrary natural number m, one can find M such that for any initial data f belonging to the closed linear span of (e_n)_{n=M}^\infty , the solution of the Cauchy problem for the heat equation decays at least at the rate t^{-m} in the sup-norm. In particular, the subspace of initial data leading to "fast" decay is finite codimensional. Moreover, the mentioned basis can be constructed as a "perturbation" of any given basis. The proof is based on a construction of bases which annihilate an infinite sequence of bounded linear functionals. We also discuss the background of the problem and possible generalizations.