Strichartz Estimates for the Vibrating Plate Equation

Details Strichartz Estimates for the Vibrating Plate Equation Strichartz Estimates for the Vibrating Plate Equation

2010-05-10

arxiv.org/abs/1005.1484v3

Mathematics

Analysis of PDEs

18 pages, 4 figures, some misprints corrected

Scientific paper

We study the dispersive properties of the linear vibrating plate (LVP) equation. Splitting it into two Schr\"odinger-type equations we show its close relation with the Schr\"odinger equation. Then, the homogeneous Sobolev spaces appear to be the natural setting to show Strichartz-type estimates for the LVP equation. By showing a Kato-Ponce inequality for homogeneous Sobolev spaces we prove the well-posedness of the Cauchy problem for the LVP equation with time-dependent potentials. Finally, we exhibit the sharpness of our results. This is achieved by finding a suitable solution for the stationary homogeneous vibrating plate equation.

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