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Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials

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  • This paper is concerned with the Cauchy problem for the nonlinear Schrödinger equations with multiple potentials. Under some assumptions on these potentials, we first obtain some sufficient conditions of blowup according to the initial energy and the Cauchy initial data directly. We next establish a sufficient condition of global existence by using potential well method and the relation between the Cauchy data and the ground state. We finally answer the question that how small the Cauchy initial data need to be for global existence by using scaling argument.
    Mathematics Subject Classification: Primary: 35A15, 35Q55; Secondary: 35B30.


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