On decay rates of the solutions of parabolic Cauchy problems. Issue 3 (June 2021)

• Record Type: Journal Article
• Authors: Bonet, José; Lusky, Wolfgang; Taskinen, Jari
• Is Part Of: Proceedings. Volume 151:Issue 3(2021)
• Page Start: 1021
• Page End: 1039
• Publication Date: 2021-06
• Subjects: Parabolic PDE, -- Cauchy problem, -- Banach space, -- Schauder basis, -- decay rate; 35K05, -- 35B40, -- 46B15, -- 46N20; Mathematics -- Periodicals; 510
• DOI: 10.1017/prm.2020.48
• Languages: English
• ISSNs: 0308-2105

Abstract

Abstract : We consider the Cauchy problem for a general class of parabolic partial differential equations in the Euclidean space ℝ N . We show that given a weighted L p -space $L_w^p({\mathbb {R}}^N)$ with 1 ⩽ p < ∞ and a fast growing weight w, there is a Schauder basis $(e_n)_{n=1}^\infty$ in $L_w^p({\mathbb {R}}^N)$ with the following property: given an arbitrary positive integer m there exists n m > 0 such that, if the initial data f belongs to the closed linear span of e n with n ⩾ n m, then the decay rate of the solution of the problem is at least t − m for large times t . The result generalizes the recent study of the authors concerning the classical linear heat equation. We present variants of the result having different methods of proofs and also consider finite polynomial decay rates instead of unlimited m .