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Weighted Strichartz Estimate for the Wave Equation and Low Regularity Solutions
Georgiev, Vladimir
D'Ancona, P.
Kubo, Hideo
2000
Abstract
In this work we study weighted Sobolev spaces in $\mathbf{R}^{n}$
generated by the Lie algebra of vector fields
\[
\left(1+\mid x\mid^{2}\right)^{1/2}\partial_{x_{j}},\; j=1,...,n.
\]
Interpolation properties and Sobolev embeddings are obtained on the
basis of a suitable localization in $\mathbf{R}^{n}$. 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 in $\left[5\right]$ and
establish global existence result for the supercritical semilinear
wave equation with non compact small initial data in these weighted
Sobolev spaces.
Series
Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
31 (2000) suppl.2
Publisher
Università degli Studi di Trieste. Dipartimento di Scienze Matematiche
Source
P. D'Ancona, V. Georgiev and H. Kubo, "Weighted Strichartz estimate for the wave equation and low regularity solutions", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 31 (2000) suppl.2, pp. 51-61.
Languages
en
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