Decay estimates for nonlocal problems via energy methods

In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form ut (x, t) = ∫Rd G (x - y) (u...

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Autor principal: Rossi, Julio Daniel
Publicado: 2009
Materias:
Energy methods
Nonlocal diffusion
p-Laplacian
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00217824_v92_n2_p163_Ignat
http://hdl.handle.net/20.500.12110/paper_00217824_v92_n2_p163_Ignat
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling paper:paper_00217824_v92_n2_p163_Ignat2023-06-08T14:42:06Z Decay estimates for nonlocal problems via energy methods Rossi, Julio Daniel Energy methods Nonlocal diffusion p-Laplacian In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form ut (x, t) = ∫Rd G (x - y) (u (y, t) - u (x, t)) d y. For example, we will consider equations like,ut (x, t) = under(∫, Rd) J (x, y) (u (y, t) - u (x, t)) d y + f (u) (x, t), and a nonlocal analogous to the p-Laplacian,ut (x, t) = under(∫, Rd) J (x, y) | u (y, t) - u (x, t) |p - 2 (u (y, t) - u (x, t)) d y . The energy method developed here allows us to obtain decay rates of the form,{norm of matrix} u (ṡ, t) {norm of matrix}Lq (Rd) ≤ C t- α, for some explicit exponent α that depends on the parameters, d, q and p, according to the problem under consideration. © 2009 Elsevier Masson SAS. All rights reserved. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2009 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00217824_v92_n2_p163_Ignat http://hdl.handle.net/20.500.12110/paper_00217824_v92_n2_p163_Ignat
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Energy methods
Nonlocal diffusion
p-Laplacian
spellingShingle Energy methods
Nonlocal diffusion
p-Laplacian
Rossi, Julio Daniel
Decay estimates for nonlocal problems via energy methods
topic_facet Energy methods
Nonlocal diffusion
p-Laplacian
description In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form ut (x, t) = ∫Rd G (x - y) (u (y, t) - u (x, t)) d y. For example, we will consider equations like,ut (x, t) = under(∫, Rd) J (x, y) (u (y, t) - u (x, t)) d y + f (u) (x, t), and a nonlocal analogous to the p-Laplacian,ut (x, t) = under(∫, Rd) J (x, y) | u (y, t) - u (x, t) |p - 2 (u (y, t) - u (x, t)) d y . The energy method developed here allows us to obtain decay rates of the form,{norm of matrix} u (ṡ, t) {norm of matrix}Lq (Rd) ≤ C t- α, for some explicit exponent α that depends on the parameters, d, q and p, according to the problem under consideration. © 2009 Elsevier Masson SAS. All rights reserved.
author Rossi, Julio Daniel
author_facet Rossi, Julio Daniel
author_sort Rossi, Julio Daniel
title Decay estimates for nonlocal problems via energy methods
title_short Decay estimates for nonlocal problems via energy methods
title_full Decay estimates for nonlocal problems via energy methods
title_fullStr Decay estimates for nonlocal problems via energy methods
title_full_unstemmed Decay estimates for nonlocal problems via energy methods
title_sort decay estimates for nonlocal problems via energy methods
publishDate 2009
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00217824_v92_n2_p163_Ignat
http://hdl.handle.net/20.500.12110/paper_00217824_v92_n2_p163_Ignat
work_keys_str_mv AT rossijuliodaniel decayestimatesfornonlocalproblemsviaenergymethods
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