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Abstract: We study the global existence of solutions in Sobolev spaces and the asymptotic behavior for the Cauchy problem of the following fourth-order Schrödinger equation with combined power-type nonlinearities , where , , , are nonzero real numbers. [Copyright &y& Elsevier]

Abstract: In this paper we study the Cauchy problem of the non-isotropically perturbed fourth-order nonlinear Schrödinger type equation: (, ), where a is a real constant, is an integer, is a nonlinear function which behaves like for some constant . 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 satisfy the conditions: , 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 . [Copyright &y& Elsevier]

Abstract: We study the global well-posedness for the Cauchy problem of the fourth-order nonlinear Schrödinger equation . By using the I-method, we prove that if , then the Cauchy problem is globally well-posed in for either the case or the case with satisfying some conditions. [Copyright &y& Elsevier]

Abstract: In this note, we establish some local and global existence results for the Cauchy problem of a class of nonlinear dispersive equations which generalize the nonlinear Schrödinger equations and the Davey–Stewartson equations. These results improve some previously obtained results by some other authors when they are restricted to certain special equations. [Copyright &y& Elsevier]

Abstract: We study the well-posedness for the Cauchy problem of the non-isotropically perturbed nonlinear Schrödinger equation , where is a real constant, is an integer, is a positive constant, and . By using Kato’s method, we establish some local and global existence results for initial data belonging to , where if , if . [Copyright &y& Elsevier]

Abstract: We study the existence of solutions for the Cauchy problem of the non-isotropically perturbed nonlinear Schrödinger equation , where , are not simultaneously vanishing real constants, is a positive constant, and . By using Kato’s method, we establish some local existence results for initial data belonging to , where if either , , or , , , if , , and if , , . Global existence is also established under some additional conditions. [Copyright &y& Elsevier]

Abstract: In this paper we establish local well-posedness in the Sobolev space with for a general class of nonlinear dispersive equations of the type , where is an elliptic differential operator on with a real symbol, is a nonlinear function which behaves like for some constant , and is a critical index suggested by a standard scaling argument. By using such local result and conservation laws, we improve the known and obtain some new global well-posedness results for the fourth-order nonlinear Schrödinger equation . [Copyright &y& Elsevier]

Abstract: We study the existence of solutions for the Cauchy problem of the non-isotropically perturbed nonlinear Schrödinger equation , where a is a real number, , α is a positive constant, and . By using Kato’s method, we establish local solvability results for initial data belonging to . [Copyright &y& Elsevier]

Abstract: In this work we study the Cauchy problem of a fourth-order nonlinear Schrödinger equation which arises from certain physical applications. We consider only the cases . Local existence of solutions for initial data belonging to Sobolev spaces with index greater than is established by using the standard contraction mapping argument. The main topic is proving that the solution is global if either the exponent of the nonlinear term is sub-critical or it is critical or super-critical but the initial data are small. This result extends the corresponding result of Fibich et al. obtained in 2002 to the super-critical case and to a more general equation. The analysis is based on applications of conservation laws for this equation. [Copyright &y& Elsevier]