Remarks on some dispersive estimates

Yonggeun Cho; Tohru Ozawa; Suxia  Xia

doi:10.3934/cpaa.2011.10.1121

Article Contents

2011, Volume 10, Issue 4: 1121-1128. Doi: 10.3934/cpaa.2011.10.1121

This issue Previous Article Regularity of solutions to an integral equation associated with Bessel potential Next Article A continuum of extinction rates for the fast diffusion equation

Remarks on some dispersive estimates

1.
Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756
2.
Department of Applied Physics, Waseda University, Tokyo, 169-8555
3.
School of Mathematics and System Sciences, Beihang University, Beijing 100191

Received: March 2010

Revised: October 2010

Published: July 2011

Abstract / Introduction Related Papers Cited by

Abstract

In this paper we consider the initial value problem for $i\partial_t u + \omega(|\nabla|) u = 0$. Under suitable smoothness and growth conditions on $\omega$, we derive dispersive estimates which is the generalization of time decay and Strichartz estimates. We unify and also simplify dispersive estimates by utilizing the Bessel function. Another main ingredient of this paper is to revisit oscillatory integrals of [2].

Keywords:

Mathematics Subject Classification: Primary: 42B37; Secondary: 35Q40.

Citation:

Related Papers

Cited by

References

[1]	J. Bergh and J. Löfström, "Interpolation Spaces," Springer-Verlag, New York, 1976.
[2]	Y. Cho and T. Ozawa, On small amplitude solutions to the generalized Boussinesq equations, Disctrete Cont. Dynam. Syst., 17 (2007), 691-711.doi: 10.3934/dcds.2007.17.691.
[3]	S. Gustafson, K. Nakanishi and T.-P. Tsai, Scattering for the Gross-Pitaevskii equation, Math. Research Letters, 13 (2006), 273-285.doi: 10.1142/S0219199709003491.
[4]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955-980.doi: 10.1353/ajm.1998.0039.
[5]	E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton Univ. Press, Princeton, N.J., 1993.