Communications on Pure and Applied Analysis (CPAA)

Remarks on some dispersive estimates

Pages: 1121 - 1128, Volume 10, Issue 4, July 2011      doi:10.3934/cpaa.2011.10.1121

       Abstract        References        Full Text (370.3K)       Related Articles       

Yonggeun Cho - Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756, South Korea (email)
Tohru Ozawa - Department of Applied Physics, Waseda University, Tokyo, 169-8555, Japan (email)
Suxia Xia - School of Mathematics and System Sciences, Beihang University, Beijing 100191, China (email)

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:  Dispersive equations, dispersive estimate, oscillatory integral, time decay, Strichartz estimate, Bessel functions.
Mathematics Subject Classification:  Primary: 42B37; Secondary: 35Q40.

Received: March 2010;      Revised: October 2010;      Available Online: April 2011.