Solutions to nonlinear higher order Schrödinger equations with small initial data on modulation spaces

Abstract

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.