Global Existence Versus Blow‐Up for the Mixed Nonlinear Schrödinger Equation

ABSTRACT This paper is concerned with blow‐up and global existence aspects for the initial‐boundary value problem to a class of mixed nonlinear Schrödinger equations. The main difficulties come from the lack of scaling invariance and the derivative terms in the nonlinearities. Some new estimates are applied to analyze the geometric structure of the energy functional and invariant manifolds, then the sharp criteria of blow‐up versus global phenomena is established. Exact sufficient conditions for the existence of global solutions are obtained under different parameter conditions, including mass‐supercritical, mass‐critical, and mass‐subcritical cases, respectively. Furthermore, the lifespan and the low blow‐up rate of solutions are established via the virial inequality.