A theorem of global existence of solutions to nonlinear wave equations in four space dimensions

Abstract In this paper, the following nonlinear wave equation is considered; □u=F(u, D u, DxD u),x∈Rn,t>0; u=u0(x), ut=u1(x), t=0. We prove that if the space dimensionn ≥ 4 and the nonlinearityF is smooth and satisfies a mild condition in a small neighborhood of the origin, then the above problem...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Gao, Jianmin [verfasserIn]

Format:	Artikel
Sprache:	Englisch

Erschienen:	1990

Schlagwörter:	Initial Data Wave Equation Mild Condition Global Solution Nonlinear Wave

Systematik:	SA 7170 SA 7170

Anmerkung:	© Springer-Verlag 1990

Übergeordnetes Werk:	Enthalten in: Monatshefte für Mathematik - Springer-Verlag, 1948, 109(1990), 2 vom: Juni, Seite 123-134
Übergeordnetes Werk:	volume:109 ; year:1990 ; number:2 ; month:06 ; pages:123-134

Links:	Volltext

DOI / URN:	10.1007/BF01302932

Katalog-ID:	OLC2059496284

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allfields	10.1007/BF01302932 doi (DE-627)OLC2059496284 (DE-He213)BF01302932-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 7170 VZ rvk SA 7170 VZ rvk Gao, Jianmin verfasserin aut A theorem of global existence of solutions to nonlinear wave equations in four space dimensions 1990 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 1990 Abstract In this paper, the following nonlinear wave equation is considered; □u=F(u, D u, DxD u),x∈Rn,t>0; u=u0(x), ut=u1(x), t=0. We prove that if the space dimensionn ≥ 4 and the nonlinearityF is smooth and satisfies a mild condition in a small neighborhood of the origin, then the above problem admits a unique and smooth global solution (in time) whenever the initial data are small and smooth. The strategy in proof is to use and improve Global Sobolev Inequalities in Minkowski space (see [8]), and to develop a generalized energy estimate for solutions. Initial Data Wave Equation Mild Condition Global Solution Nonlinear Wave Enthalten in Monatshefte für Mathematik Springer-Verlag, 1948 109(1990), 2 vom: Juni, Seite 123-134 (DE-627)129492191 (DE-600)206474-1 (DE-576)014887657 0026-9255 nnns volume:109 year:1990 number:2 month:06 pages:123-134 https://doi.org/10.1007/BF01302932 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_22 GBV_ILN_30 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_65 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2020 GBV_ILN_2088 GBV_ILN_2333 GBV_ILN_4012 GBV_ILN_4029 GBV_ILN_4082 GBV_ILN_4103 GBV_ILN_4126 GBV_ILN_4193 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4311 GBV_ILN_4314 GBV_ILN_4315 GBV_ILN_4316 GBV_ILN_4317 GBV_ILN_4318 GBV_ILN_4325 GBV_ILN_4700 SA 7170 SA 7170 AR 109 1990 2 06 123-134
spelling	10.1007/BF01302932 doi (DE-627)OLC2059496284 (DE-He213)BF01302932-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 7170 VZ rvk SA 7170 VZ rvk Gao, Jianmin verfasserin aut A theorem of global existence of solutions to nonlinear wave equations in four space dimensions 1990 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 1990 Abstract In this paper, the following nonlinear wave equation is considered; □u=F(u, D u, DxD u),x∈Rn,t>0; u=u0(x), ut=u1(x), t=0. We prove that if the space dimensionn ≥ 4 and the nonlinearityF is smooth and satisfies a mild condition in a small neighborhood of the origin, then the above problem admits a unique and smooth global solution (in time) whenever the initial data are small and smooth. The strategy in proof is to use and improve Global Sobolev Inequalities in Minkowski space (see [8]), and to develop a generalized energy estimate for solutions. Initial Data Wave Equation Mild Condition Global Solution Nonlinear Wave Enthalten in Monatshefte für Mathematik Springer-Verlag, 1948 109(1990), 2 vom: Juni, Seite 123-134 (DE-627)129492191 (DE-600)206474-1 (DE-576)014887657 0026-9255 nnns volume:109 year:1990 number:2 month:06 pages:123-134 https://doi.org/10.1007/BF01302932 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_22 GBV_ILN_30 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_65 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2020 GBV_ILN_2088 GBV_ILN_2333 GBV_ILN_4012 GBV_ILN_4029 GBV_ILN_4082 GBV_ILN_4103 GBV_ILN_4126 GBV_ILN_4193 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4311 GBV_ILN_4314 GBV_ILN_4315 GBV_ILN_4316 GBV_ILN_4317 GBV_ILN_4318 GBV_ILN_4325 GBV_ILN_4700 SA 7170 SA 7170 AR 109 1990 2 06 123-134
allfields_unstemmed	10.1007/BF01302932 doi (DE-627)OLC2059496284 (DE-He213)BF01302932-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 7170 VZ rvk SA 7170 VZ rvk Gao, Jianmin verfasserin aut A theorem of global existence of solutions to nonlinear wave equations in four space dimensions 1990 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 1990 Abstract In this paper, the following nonlinear wave equation is considered; □u=F(u, D u, DxD u),x∈Rn,t>0; u=u0(x), ut=u1(x), t=0. We prove that if the space dimensionn ≥ 4 and the nonlinearityF is smooth and satisfies a mild condition in a small neighborhood of the origin, then the above problem admits a unique and smooth global solution (in time) whenever the initial data are small and smooth. The strategy in proof is to use and improve Global Sobolev Inequalities in Minkowski space (see [8]), and to develop a generalized energy estimate for solutions. Initial Data Wave Equation Mild Condition Global Solution Nonlinear Wave Enthalten in Monatshefte für Mathematik Springer-Verlag, 1948 109(1990), 2 vom: Juni, Seite 123-134 (DE-627)129492191 (DE-600)206474-1 (DE-576)014887657 0026-9255 nnns volume:109 year:1990 number:2 month:06 pages:123-134 https://doi.org/10.1007/BF01302932 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_22 GBV_ILN_30 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_65 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2020 GBV_ILN_2088 GBV_ILN_2333 GBV_ILN_4012 GBV_ILN_4029 GBV_ILN_4082 GBV_ILN_4103 GBV_ILN_4126 GBV_ILN_4193 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4311 GBV_ILN_4314 GBV_ILN_4315 GBV_ILN_4316 GBV_ILN_4317 GBV_ILN_4318 GBV_ILN_4325 GBV_ILN_4700 SA 7170 SA 7170 AR 109 1990 2 06 123-134
allfieldsGer	10.1007/BF01302932 doi (DE-627)OLC2059496284 (DE-He213)BF01302932-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 7170 VZ rvk SA 7170 VZ rvk Gao, Jianmin verfasserin aut A theorem of global existence of solutions to nonlinear wave equations in four space dimensions 1990 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 1990 Abstract In this paper, the following nonlinear wave equation is considered; □u=F(u, D u, DxD u),x∈Rn,t>0; u=u0(x), ut=u1(x), t=0. We prove that if the space dimensionn ≥ 4 and the nonlinearityF is smooth and satisfies a mild condition in a small neighborhood of the origin, then the above problem admits a unique and smooth global solution (in time) whenever the initial data are small and smooth. The strategy in proof is to use and improve Global Sobolev Inequalities in Minkowski space (see [8]), and to develop a generalized energy estimate for solutions. Initial Data Wave Equation Mild Condition Global Solution Nonlinear Wave Enthalten in Monatshefte für Mathematik Springer-Verlag, 1948 109(1990), 2 vom: Juni, Seite 123-134 (DE-627)129492191 (DE-600)206474-1 (DE-576)014887657 0026-9255 nnns volume:109 year:1990 number:2 month:06 pages:123-134 https://doi.org/10.1007/BF01302932 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_22 GBV_ILN_30 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_65 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2020 GBV_ILN_2088 GBV_ILN_2333 GBV_ILN_4012 GBV_ILN_4029 GBV_ILN_4082 GBV_ILN_4103 GBV_ILN_4126 GBV_ILN_4193 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4311 GBV_ILN_4314 GBV_ILN_4315 GBV_ILN_4316 GBV_ILN_4317 GBV_ILN_4318 GBV_ILN_4325 GBV_ILN_4700 SA 7170 SA 7170 AR 109 1990 2 06 123-134
allfieldsSound	10.1007/BF01302932 doi (DE-627)OLC2059496284 (DE-He213)BF01302932-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 7170 VZ rvk SA 7170 VZ rvk Gao, Jianmin verfasserin aut A theorem of global existence of solutions to nonlinear wave equations in four space dimensions 1990 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 1990 Abstract In this paper, the following nonlinear wave equation is considered; □u=F(u, D u, DxD u),x∈Rn,t>0; u=u0(x), ut=u1(x), t=0. We prove that if the space dimensionn ≥ 4 and the nonlinearityF is smooth and satisfies a mild condition in a small neighborhood of the origin, then the above problem admits a unique and smooth global solution (in time) whenever the initial data are small and smooth. The strategy in proof is to use and improve Global Sobolev Inequalities in Minkowski space (see [8]), and to develop a generalized energy estimate for solutions. Initial Data Wave Equation Mild Condition Global Solution Nonlinear Wave Enthalten in Monatshefte für Mathematik Springer-Verlag, 1948 109(1990), 2 vom: Juni, Seite 123-134 (DE-627)129492191 (DE-600)206474-1 (DE-576)014887657 0026-9255 nnns volume:109 year:1990 number:2 month:06 pages:123-134 https://doi.org/10.1007/BF01302932 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_22 GBV_ILN_30 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_65 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2020 GBV_ILN_2088 GBV_ILN_2333 GBV_ILN_4012 GBV_ILN_4029 GBV_ILN_4082 GBV_ILN_4103 GBV_ILN_4126 GBV_ILN_4193 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4311 GBV_ILN_4314 GBV_ILN_4315 GBV_ILN_4316 GBV_ILN_4317 GBV_ILN_4318 GBV_ILN_4325 GBV_ILN_4700 SA 7170 SA 7170 AR 109 1990 2 06 123-134
language	English
source	Enthalten in Monatshefte für Mathematik 109(1990), 2 vom: Juni, Seite 123-134 volume:109 year:1990 number:2 month:06 pages:123-134
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author	Gao, Jianmin
spellingShingle	Gao, Jianmin ddc 510 ssgn 17,1 rvk SA 7170 misc Initial Data misc Wave Equation misc Mild Condition misc Global Solution misc Nonlinear Wave A theorem of global existence of solutions to nonlinear wave equations in four space dimensions
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topic_title	510 VZ 17,1 ssgn SA 7170 VZ rvk A theorem of global existence of solutions to nonlinear wave equations in four space dimensions Initial Data Wave Equation Mild Condition Global Solution Nonlinear Wave
topic	ddc 510 ssgn 17,1 rvk SA 7170 misc Initial Data misc Wave Equation misc Mild Condition misc Global Solution misc Nonlinear Wave
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title	A theorem of global existence of solutions to nonlinear wave equations in four space dimensions
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title_full	A theorem of global existence of solutions to nonlinear wave equations in four space dimensions
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title_sort	a theorem of global existence of solutions to nonlinear wave equations in four space dimensions
title_auth	A theorem of global existence of solutions to nonlinear wave equations in four space dimensions
abstract	Abstract In this paper, the following nonlinear wave equation is considered; □u=F(u, D u, DxD u),x∈Rn,t>0; u=u0(x), ut=u1(x), t=0. We prove that if the space dimensionn ≥ 4 and the nonlinearityF is smooth and satisfies a mild condition in a small neighborhood of the origin, then the above problem admits a unique and smooth global solution (in time) whenever the initial data are small and smooth. The strategy in proof is to use and improve Global Sobolev Inequalities in Minkowski space (see [8]), and to develop a generalized energy estimate for solutions. © Springer-Verlag 1990
abstractGer	Abstract In this paper, the following nonlinear wave equation is considered; □u=F(u, D u, DxD u),x∈Rn,t>0; u=u0(x), ut=u1(x), t=0. We prove that if the space dimensionn ≥ 4 and the nonlinearityF is smooth and satisfies a mild condition in a small neighborhood of the origin, then the above problem admits a unique and smooth global solution (in time) whenever the initial data are small and smooth. The strategy in proof is to use and improve Global Sobolev Inequalities in Minkowski space (see [8]), and to develop a generalized energy estimate for solutions. © Springer-Verlag 1990
abstract_unstemmed	Abstract In this paper, the following nonlinear wave equation is considered; □u=F(u, D u, DxD u),x∈Rn,t>0; u=u0(x), ut=u1(x), t=0. We prove that if the space dimensionn ≥ 4 and the nonlinearityF is smooth and satisfies a mild condition in a small neighborhood of the origin, then the above problem admits a unique and smooth global solution (in time) whenever the initial data are small and smooth. The strategy in proof is to use and improve Global Sobolev Inequalities in Minkowski space (see [8]), and to develop a generalized energy estimate for solutions. © Springer-Verlag 1990
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title_short	A theorem of global existence of solutions to nonlinear wave equations in four space dimensions
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up_date	2024-07-03T22:19:00.122Z
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