Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 83, pp. 1-22.

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

Authors: Runzhang Xu (Harbin Engineering Univ., Harbin, China)
         Qiang Lin  (Harbin Engineering Univ., Harbin, China)
         Shaohua Chen (Cape Breton Univ., Sydney, NS, Canada)
         Guojun Wen  (Harbin Engineering Univ., Harbin, China)
         Wei Lian    (Harbin Engineering Univ., Harbin, China)

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.

Submitted January 11, 2019. Published June 24, 2019.
Math Subject Classifications: 35B44, 35G25, 35A01, 35Q55.
Key Words: Fourth-order Schrodinger equation; global solution; blowup; 
           variational problem; invariant manifolds.