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This paper investigates the local and global existence for the inhomogeneous nonlinear Schrödinger equation with the nonlinearity λ|x|^{-b}|u|^{β}u. It is show that a global solution exists in the mass-subcritical for large data in the spaces L^{p}, p < 2 under some suitable conditions on b,β and p. The solution is established using a data-decomposition argument, two kinds of generalized Strichartz estimates in Lorentz spaces and a interpolation theorem.

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This paper is concerned with the Cauchy problem for the nonlinear fourth-order Schrödinger equation on R^{n}, with the nonlinearity of Hartree-type (| ·|^{-γ}∗|u|^{2} )u .It is shown that a global solution exists for initial data in the spaces L^{p} (p < 2) under somesuitable conditions on γ, n and p. The solution is established by using a data-decomposition argument, two kinds of generalized Strichartz estimates and a interpolation theorem.

This paper investigates the local and global existence for the inhomogeneous nonlinear Schrödinger equation with the nonlinearity 𝜆|x| −b |u| 𝛽 u. It is show that a global solution exists in the mass-subcritical for large data in the spaces L p , p < 2 under some suitable conditions on b, 𝛽, and p. The solution is established using a data-decomposition argument, two kinds of generalized Strichartz estimates in Lorentz spaces and a interpolation theorem.