Difficulties in obtaining finite time blowup for fourth-order semilinear Schrödinger equations in the variational method frame

Abstract

This article concerns the Cauchy problem for fourth-order semilinear Schrodinger equations. By constructing a variational problem and some invariant manifolds, we prove the existence of a global solution. Then we analyze the difficulties in proving the finite time blowup of the solution for the corresponding problem in the frame of the variational method. Understanding the finite time blowup of solutions, without radial initial data, still remains an open problem.