Solutions to nonlinear higher order Schrödinger equations with small initial data on modulation spaces.(English)Zbl 1375.35493

Summary: In this paper, we consider the Cauchy problem for the nonlinear higher order Schrödinger equations on modulation spaces $$M_{p,q}^s$$ and show the existence of a unique global solution by using integrability of time decay factors of time decay estimates. As a result, we are able to deal with wider classes of a nonlinearity and a solution space. Moreover, we study time decay estimates of a semi-group $$e^{it\phi(\sqrt{-\Delta})}$$ with a polynomial symbol $$\phi$$. Considering multiplicities of critical points and inflection points of $$\phi$$ carefully, we have time decay estimates with better time decay rate.

MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35G25 Initial value problems for nonlinear higher-order PDEs 35G05 Linear higher-order PDEs
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