Weighted decay estimates for the wave equation

Piero D'Ancona; Vladimir Georgiev; Hideo Kubo

doi:10.1006/jdeq.2000.3983

Weighted decay estimates for the wave equation

Piero D'Ancona, Vladimir Georgiev, Hideo Kubo

Research output: Contribution to journal › Article › peer-review

35 Citations (Scopus)

Abstract

In this work we study weighted Sobolev spaces in Rⁿ generated by the Lie algebra of vector fields (1 + x ²)^1/2 ∂_xj, j = 1, ..., n. Interpolation properties and Sobolev embeddings are obtained on the basis of a suitable localization in Rⁿ. As an application we derive weighted L^q estimates for the solution of the homogeneous wave equation. For the inhomogeneous wave equation we generalize the weighted Strichartz estimate established by V. Georgiev (1997, Amer. J. Math. 119, 1291-1319) and establish global existence results for the supercritical semilinear wave equation with non-compact small initial data in these weighted Sobolev spaces.

Original language	English
Pages (from-to)	146-208
Number of pages	63
Journal	Journal of Differential Equations
Volume	177
Issue number	1
DOIs	https://doi.org/10.1006/jdeq.2000.3983
Publication status	Published - 2001 Nov 20
Externally published	Yes

Keywords

Decay estimates
Global solution
Semilinear equation
Supercritical
Wave equation
Weighted sobolev spaces

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1006/jdeq.2000.3983

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title = "Weighted decay estimates for the wave equation",

abstract = "In this work we study weighted Sobolev spaces in Rn generated by the Lie algebra of vector fields (1 + x 2)1/2 ∂xj, j = 1, ..., n. Interpolation properties and Sobolev embeddings are obtained on the basis of a suitable localization in Rn. As an application we derive weighted Lq estimates for the solution of the homogeneous wave equation. For the inhomogeneous wave equation we generalize the weighted Strichartz estimate established by V. Georgiev (1997, Amer. J. Math. 119, 1291-1319) and establish global existence results for the supercritical semilinear wave equation with non-compact small initial data in these weighted Sobolev spaces.",

keywords = "Decay estimates, Global solution, Semilinear equation, Supercritical, Wave equation, Weighted sobolev spaces",

author = "Piero D'Ancona and Vladimir Georgiev and Hideo Kubo",

note = "Funding Information: 1The first and second authors were partially supported by the Programma Nazionale M.U.R.S.T. {\textquoteleft}{\textquoteleft}Problemi e Metodi nella Teoria delle Equazioni Iperboliche.{\textquoteright}{\textquoteright} The third author was partially supported by Grant-in-Aid for Science Research (No. 09304012), Ministry of Education, Science and Culture, Japan.",

year = "2001",

month = nov,

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doi = "10.1006/jdeq.2000.3983",

language = "English",

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journal = "Journal of Differential Equations",

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T1 - Weighted decay estimates for the wave equation

AU - D'Ancona, Piero

AU - Georgiev, Vladimir

AU - Kubo, Hideo

N1 - Funding Information: 1The first and second authors were partially supported by the Programma Nazionale M.U.R.S.T. ‘‘Problemi e Metodi nella Teoria delle Equazioni Iperboliche.’’ The third author was partially supported by Grant-in-Aid for Science Research (No. 09304012), Ministry of Education, Science and Culture, Japan.

PY - 2001/11/20

Y1 - 2001/11/20

N2 - In this work we study weighted Sobolev spaces in Rn generated by the Lie algebra of vector fields (1 + x 2)1/2 ∂xj, j = 1, ..., n. Interpolation properties and Sobolev embeddings are obtained on the basis of a suitable localization in Rn. As an application we derive weighted Lq estimates for the solution of the homogeneous wave equation. For the inhomogeneous wave equation we generalize the weighted Strichartz estimate established by V. Georgiev (1997, Amer. J. Math. 119, 1291-1319) and establish global existence results for the supercritical semilinear wave equation with non-compact small initial data in these weighted Sobolev spaces.

AB - In this work we study weighted Sobolev spaces in Rn generated by the Lie algebra of vector fields (1 + x 2)1/2 ∂xj, j = 1, ..., n. Interpolation properties and Sobolev embeddings are obtained on the basis of a suitable localization in Rn. As an application we derive weighted Lq estimates for the solution of the homogeneous wave equation. For the inhomogeneous wave equation we generalize the weighted Strichartz estimate established by V. Georgiev (1997, Amer. J. Math. 119, 1291-1319) and establish global existence results for the supercritical semilinear wave equation with non-compact small initial data in these weighted Sobolev spaces.

KW - Decay estimates

KW - Global solution

KW - Semilinear equation

KW - Supercritical

KW - Wave equation

KW - Weighted sobolev spaces

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