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.
| Original language | English |
|---|---|
| Pages (from-to) | 146-208 |
| Number of pages | 63 |
| Journal | Journal of Differential Equations |
| Volume | 177 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2001 Nov 20 |
| Externally published | Yes |
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Keywords
- Decay estimates
- Global solution
- Semilinear equation
- Supercritical
- Wave equation
- Weighted sobolev spaces
ASJC Scopus subject areas
- Analysis
Cite this
Weighted decay estimates for the wave equation. / D'Ancona, Piero; Gueorguiev, Vladimir Simeonov; Kubo, Hideo.
In: Journal of Differential Equations, Vol. 177, No. 1, 20.11.2001, p. 146-208.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Weighted decay estimates for the wave equation
AU - D'Ancona, Piero
AU - Gueorguiev, Vladimir Simeonov
AU - Kubo, Hideo
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
UR - http://www.scopus.com/inward/record.url?scp=0035923674&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0035923674&partnerID=8YFLogxK
U2 - 10.1006/jdeq.2000.3983
DO - 10.1006/jdeq.2000.3983
M3 - Article
VL - 177
SP - 146
EP - 208
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
IS - 1
ER -