Well-posedness of the fourth-order perturbed Schrödinger type equation in non-isotropic Sobolev spaces |
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Authors: | Xiangqing Zhao Cuihua Guo Wancheng Sheng |
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Institution: | a Department of Mathematics, Zhejiang Ocean University, Zhoushan, Zhejiang 316000, China b School of Mathematical Science, Shanxi University, Taiyuan, Shanxi 030006, China c Department of Mathematics, Shanghai University, Shanghai 200436, China d Faculty of Applied Mathematics, Guangdong University of Technology, Guangzhou, Guangdong 510090, China |
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Abstract: | In this paper we study the Cauchy problem of the non-isotropically perturbed fourth-order nonlinear Schrödinger type equation: ((x1,x2,…,xn)∈Rn, t?0), where a is a real constant, 1?d<n is an integer, g(x,|u|)u is a nonlinear function which behaves like α|u|u for some constant α>0. By using Kato method, we prove that this perturbed fourth-order Schrödinger type equation is locally well-posed with initial data belonging to the non-isotropic Sobolev spaces provided that s1,s2 satisfy the conditions: s1?0, s2?0 for or for with some additional conditions. Furthermore, by using non-isotropic Sobolev inequality and energy method, we obtain some global well-posedness results for initial data belonging to non-isotropic Sobolev spaces . |
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Keywords: | Schrö dinger type equation Cauchy problem Kato method Non-isotropic Sobolev space |
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