Estimates for wave and Klein-Gordon equations on modulation spaces |
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Authors: | JieCheng Chen DaShan Fan |
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Institution: | 1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China 2. Department of Mathematics, University of Wisconsin-Milwaukee, Milwaukee, WI, 53201, USA
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Abstract: | We prove that the fundamental semi-group $e^{it\left( {m^2 I + \left| \Delta \right|} \right)^{1/2} }$ (m ?? 0) of the Klein-Gordon equation is bounded on the modulation space M p,q s (? n ) for all 0 < p, q ? ?? and s ?? ?. Similarly, we prove that the wave semi-group $e^{it\left| \Delta \right|^{1/2} }$ is bounded on the Hardy type modulation spaces ?? p,q s (? n ) for all 0 < p, q ? ??, and s ?? ?. All the bounds have an asymptotic factor t n|1/p?1/2| as t goes to the infinity. These results extend some known results for the case of p ? 1. Also, some applications for the Cauchy problems related to the semi-group $e^{it\left( {m^2 I + \left| \Delta \right|} \right)^{1/2} }$ are obtained. Finally we discuss the optimum of the factor t n|1/p?1/2| and raise some unsolved problems. |
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Keywords: | Klein-Gordon equation wave equation modulation space |
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