Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line

Kamin and Vázquez [11] proved in 1991 that solutions to the Cauchy-Dirichlet problem for the porous medium equation ut = (um)xx, m > 1, on the half-line with zero boundary data and nonnegative compactly supported integrable initial data behave for large times as a dipole-type solution to the...

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Detalles Bibliográficos
Autor principal: Wolanski, Noemi Irene
Publicado: 2017
Materias:
Asymptotic Behavior
Matched Asymptotics
Porous Medium Equation on the Half-Line
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_15361365_v17_n2_p245_Cortazar
http://hdl.handle.net/20.500.12110/paper_15361365_v17_n2_p245_Cortazar
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_15361365_v17_n2_p245_Cortazar
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spelling paper:paper_15361365_v17_n2_p245_Cortazar2023-06-08T16:20:10Z Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line Wolanski, Noemi Irene Asymptotic Behavior Matched Asymptotics Porous Medium Equation on the Half-Line Kamin and Vázquez [11] proved in 1991 that solutions to the Cauchy-Dirichlet problem for the porous medium equation ut = (um)xx, m > 1, on the half-line with zero boundary data and nonnegative compactly supported integrable initial data behave for large times as a dipole-type solution to the equation having the same first moment as the initial data, with an error which is o(t-1/m). However, on sets of the form 0 < x < g(t), with g(t) = o(t1/(2m)) as t →, in the so-called near field, a scale which includes the particular case of compact sets, the dipole solution is o( t-1/m), and their result gives neither the right rate of decay of the solution nor a nontrivial asymptotic profile. In this paper, we will improve the estimate for the error, showing that it is o(t-(2m+1)/(2m2) (1+x)1/m). This allows in particular to obtain a nontrivial asymptotic profile in the near field limit, which is a multiple of x1/m, thus improving in this scale the results of Kamin and Vázquez. © 2017 by De Gruyter. Fil:Wolanski, N. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2017 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_15361365_v17_n2_p245_Cortazar http://hdl.handle.net/20.500.12110/paper_15361365_v17_n2_p245_Cortazar
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Asymptotic Behavior
Matched Asymptotics
Porous Medium Equation on the Half-Line
spellingShingle Asymptotic Behavior
Matched Asymptotics
Porous Medium Equation on the Half-Line
Wolanski, Noemi Irene
Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line
topic_facet Asymptotic Behavior
Matched Asymptotics
Porous Medium Equation on the Half-Line
description Kamin and Vázquez [11] proved in 1991 that solutions to the Cauchy-Dirichlet problem for the porous medium equation ut = (um)xx, m > 1, on the half-line with zero boundary data and nonnegative compactly supported integrable initial data behave for large times as a dipole-type solution to the equation having the same first moment as the initial data, with an error which is o(t-1/m). However, on sets of the form 0 < x < g(t), with g(t) = o(t1/(2m)) as t →, in the so-called near field, a scale which includes the particular case of compact sets, the dipole solution is o( t-1/m), and their result gives neither the right rate of decay of the solution nor a nontrivial asymptotic profile. In this paper, we will improve the estimate for the error, showing that it is o(t-(2m+1)/(2m2) (1+x)1/m). This allows in particular to obtain a nontrivial asymptotic profile in the near field limit, which is a multiple of x1/m, thus improving in this scale the results of Kamin and Vázquez. © 2017 by De Gruyter.
author Wolanski, Noemi Irene
author_facet Wolanski, Noemi Irene
author_sort Wolanski, Noemi Irene
title Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line
title_short Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line
title_full Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line
title_fullStr Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line
title_full_unstemmed Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line
title_sort near field asymptotic behavior for the porous medium equation on the half-line
publishDate 2017
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_15361365_v17_n2_p245_Cortazar
http://hdl.handle.net/20.500.12110/paper_15361365_v17_n2_p245_Cortazar
work_keys_str_mv AT wolanskinoemiirene nearfieldasymptoticbehaviorfortheporousmediumequationonthehalfline
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