Critical exponents for a semilinear parabolic equation with variable reaction

We study the blow-up phenomenon for non-negative solutions to the following parabolic problem: [equation presented] where 0 < p = min p p(x) max p = p+ is a smooth bounded function. After discussing existence and uniqueness, we characterize the critical exponents for this problem. We prove th...

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Autor principal: Rossi, Julio Daniel
Publicado: 2012
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03082105_v142A_n5_p1027_Ferreira
http://hdl.handle.net/20.500.12110/paper_03082105_v142A_n5_p1027_Ferreira
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_03082105_v142A_n5_p1027_Ferreira
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spelling paper:paper_03082105_v142A_n5_p1027_Ferreira2023-06-08T15:31:36Z Critical exponents for a semilinear parabolic equation with variable reaction Rossi, Julio Daniel We study the blow-up phenomenon for non-negative solutions to the following parabolic problem: [equation presented] where 0 < p = min p p(x) max p = p+ is a smooth bounded function. After discussing existence and uniqueness, we characterize the critical exponents for this problem. We prove that there are solutions with blow-up in finite time if and only if p+ > 1. When ω = ℝ N we show that if p > 1 + 2/N, then there are global non-trivial solutions, while if 1 < p p+ 1 + 2/N, then all solutions to the problem blow up in finite time. Moreover, in the case when p < 1 + 2/N < p+, there are functions p(x) such that all solutions blow up in finite time and functions p(x) such that the problem possesses global non-trivial solutions. When ω is a bounded domain we prove that there are functions p(x) and domains ω such that all solutions to the problem blow up in finite time. On the other hand, if ω is small enough, then the problem possesses global non-trivial solutions regardless of the size of p(x). © 2012 Royal Society of Edinburgh. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2012 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03082105_v142A_n5_p1027_Ferreira http://hdl.handle.net/20.500.12110/paper_03082105_v142A_n5_p1027_Ferreira
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
description We study the blow-up phenomenon for non-negative solutions to the following parabolic problem: [equation presented] where 0 < p = min p p(x) max p = p+ is a smooth bounded function. After discussing existence and uniqueness, we characterize the critical exponents for this problem. We prove that there are solutions with blow-up in finite time if and only if p+ > 1. When ω = ℝ N we show that if p > 1 + 2/N, then there are global non-trivial solutions, while if 1 < p p+ 1 + 2/N, then all solutions to the problem blow up in finite time. Moreover, in the case when p < 1 + 2/N < p+, there are functions p(x) such that all solutions blow up in finite time and functions p(x) such that the problem possesses global non-trivial solutions. When ω is a bounded domain we prove that there are functions p(x) and domains ω such that all solutions to the problem blow up in finite time. On the other hand, if ω is small enough, then the problem possesses global non-trivial solutions regardless of the size of p(x). © 2012 Royal Society of Edinburgh.
author Rossi, Julio Daniel
spellingShingle Rossi, Julio Daniel
Critical exponents for a semilinear parabolic equation with variable reaction
author_facet Rossi, Julio Daniel
author_sort Rossi, Julio Daniel
title Critical exponents for a semilinear parabolic equation with variable reaction
title_short Critical exponents for a semilinear parabolic equation with variable reaction
title_full Critical exponents for a semilinear parabolic equation with variable reaction
title_fullStr Critical exponents for a semilinear parabolic equation with variable reaction
title_full_unstemmed Critical exponents for a semilinear parabolic equation with variable reaction
title_sort critical exponents for a semilinear parabolic equation with variable reaction
publishDate 2012
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03082105_v142A_n5_p1027_Ferreira
http://hdl.handle.net/20.500.12110/paper_03082105_v142A_n5_p1027_Ferreira
work_keys_str_mv AT rossijuliodaniel criticalexponentsforasemilinearparabolicequationwithvariablereaction
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