# On nonlinear Schrödinger equations, II.*H*^{S}-solutions and unconditional well-posedness

## Authors

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DOI: 10.1007/BF02787794

- Cite this article as:
- Kato, T. J. Anal. Math. (1995) 67: 281. doi:10.1007/BF02787794

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## Abstract

We consider the nonlinear Schrödinger equation (NLS) (see below) with a general “potential”*F(u)*, for which there are in general no conservation laws. The main assumption on*F(u)* is a growth rate*O(|u|*^{k}) for large |*u*|, in addition to some smoothness depending on the problem considered. A uniqueness theorem is proved with minimal smoothness assumption on*F* and*u*, which is useful in eliminating the “auxiliary conditions” in many cases. A new local existence theorem for*H*^{S}-solutions is proved using an auxiliary space of Lebesgue type (rather than Besov type); here the main assumption is that*k*≤1+4/(*m−2s*) if*s<m*/2,*k*<∞ if*s=m*/2 (no assumption if*s>m*/2). Moreover, a general existence theorem is proved for global*H*^{S}-solutions with small initial data, under the main additional condition that*F(u)*=*O*(|u|^{1+4/m}) for small |*u*|; in particular*F(u)* need not be (quasi-) homogeneous or in the critical case. The results are valid for all*s*≥0 if*m*≤6; there are some restrictions if*m*≥7 and if*F(u)* is*not* a polynomial in*u* and\(\bar u\).