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.
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Acknowledgements
Open access funding provided by University of Helsinki including Helsinki University Central Hospital. The authors would like to thank Thierry Gallay (Grenoble) for discussions which helped in the final formulation of our results. The research of Bonet was partially supported by the projects MTM2016-76647-P and GV Prometeo 2017/102. The research of Taskinen was partially supported by the research grant from the Faculty of Science of the University of Helsinki.
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Bonet, J., Lusky, W. & Taskinen, J. Schauder bases and the decay rate of the heat equation. J. Evol. Equ. 19, 717–728 (2019). https://doi.org/10.1007/s00028-019-00492-x
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DOI: https://doi.org/10.1007/s00028-019-00492-x