Local well-posedness for the inhomogeneous nonlinear Schrödinger equation

Abstract

We consider the Cauchy problem for the inhomogeneous nonlinear SchrödingerOnly partial results are known for the local existence in the subcritical case α < (4 − 2b)/(N − 2s) and much more less in the critical case α = (4 − 2b)/(N − 2s). In this paper, we develop a local well-posedness theory for the both cases. In particular, we establish new results for the continuous dependence and for the unconditional uniqueness. Our approach provides simple proofs and allows us to obtain lower bounds of the blowup rate and of the life span. The Lorentz spaces and the Strichartz estimates play important roles in our argument. In particular this enables us to reach the critical case and to unify results for b = 0 and b > 0.

Long time dynamics for the focusing nonlinear Schrödinger equation with exponential nonlinearities

“…In the case α > 1, we take 1 n = 0 or n = ∞. In the case α ≤ 1, we take 1 n = 1−α 2 or n = 2 1−α . The first two conditions in (4.31) are satisfied.…”

Global existence and scattering for the inhomogeneous nonlinear Schrödinger equation

“…In this paper, which is a continuation of our previous article [1], we investigate the global existence and the asymptotic behavior for the inhomogeneous nonlinear Schrödinger equation…”

“…See for example [23,Section 6]. The local theory for (1.1) has been established in [1,15,20,24,26,29,30] under the conditions (K 1 ) − (K 2 ) below. We mention also that the study of the standing waves for (1.1) is done in [4,22,32].…”

“…Our aim here is to improve this result in terms of the allowed values of α and on the smallness of initial data similar to the case b = 0 in [10]. To do this we exploit the scaling of the equation and the blow up criterion given in [1] and we establish the Strichartz estimates for non-admissible pairs to handle the potential K.…”

“…In fact, if u is a solution of (1. 1) with K(x) = |x| −b , then for all λ > 0, u λ is also a solution of (1.1), where…”