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 |
Keywords
- Decay estimates
- Global solution
- Semilinear equation
- Supercritical
- Wave equation
- Weighted sobolev spaces
ASJC Scopus subject areas
- Analysis
- Applied Mathematics
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Cite this
D'Ancona, P., Georgiev, V., & Kubo, H. (2001). Weighted decay estimates for the wave equation. Journal of Differential Equations, 177(1), 146-208. https://doi.org/10.1006/jdeq.2000.3983