2026-06-28T15:10:49Zhttps://docta.ucm.es/rest/oai/requestoai:docta.ucm.es:20.500.14352/886992024-10-31T15:13:36Zcom_20.500.14352_14col_20.500.14352_15
The heat flow in an optimal Fréchet space of unbounded initial data in Rd
Rodríguez Bernal, Aníbal
Robinson, James C.
In this paper we show that solutions of the heat equation that are given in terms of the heat kernel define semigroups on the family of Fréchet spaces Lp0 (Rd ), the intersection (over all ε > 0) of the spaces Lpε (Rd ) of functions such that ∫ Rd e−ε|x|2 |f (x)|p dx < ∞. These spaces consist of functions that are ‘large at infinity’, and L10 (Rd ) is the maximal space in which one can use the heat kernel to obtain globally-defined solutions of the heat equation. We prove suitable estimates from Lp0 (Rd ) into Lq0 (Rd ), q ≥ p, for these semigroups. We then consider the heat semigroup posed in spaces that are dual to these spaces of functions, namely the spaces Lp−ε (Rd ) of very-rapidly decreasing functions such that ∫ Rd eε|x|2 |f (x)|p dx < ∞. We show that (Lppε (Rd ))' = Lq−qε (Rd ) (with 1 <p< ∞ and (p, q) conjugate), and that the heat flow on Lpε (Rd ) is the adjoint of the flow on Lq−δ (Rd ) for an appropriate (time-dependent) choice of δ.
2023-11-13T17:16:58Z
2023-11-13T17:16:58Z
2023-11-13T17:16:58Z
2020-07-15
journal article
https://hdl.handle.net/20.500.14352/88699
XXXX-XXXX
10.1016/j.jde.2020.07.017
eng
info:eu-repo/grantAgreement/UCM/Ecuaciones en derivadas parciales: dinámica asintótica y perturbaciones/MTM2016-75465-P
info:eu-repo/grantAgreement/UCM/Aspectos lineales y no lineales en ecuaciones en derivadas parciales. Dinámica asintótica y perturbaciones/PID2019-103860GB-I00
info:eu-repo/grantAgreement/MCIN//PRX17/00522
info:eu-repo/grantAgreement/MINECO//SEV-2015-0554/ES/INSTITUTO DE CIENCIAS MATEMATICAS/
EP/R023778/1
GR58/08
Robinson, J. C., & Rodríguez-Bernal, A. (2020). The heat flow in an optimal Fréchet space of unbounded initial data in Rd. Journal Of Differential Equations, 269(11), 10277-10321. https://doi.org/10.1016/j.jde.2020.07.017
http://creativecommons.org/licenses/by-nc-nd/4.0/
open access
Attribution-NonCommercial-NoDerivatives 4.0 International
Elsevier