Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient

Abstract

We consider the fourth-order Schrödinger equationwhere α > 0, µ = ±1 or 0 and λ ∈ C. Firstly, we prove local well-posedness in H 4 R N in both H 4 subcritical and critical case: α > 0, (N − 8)α ≤ 8. Then, for any given compact set K ⊂ R N , we construct H 4 (R N ) solutions that are defined on (−T, 0) for some T > 0, and blow up exactly on K at t = 0.

Global solutions for $H^s$-critical nonlinear biharmonic Schrödinger equation

“…Since then, the local and global well-posedness for (1.1) have been widely studied in recent years. See [5,6,8,10,11,14,16,20,22,24] and references therein.…”

“…We note that the power type nonlinearity f (u) = λ |u| α u or f (u) = λ |u| α+1 with λ ∈ R, α > 0 is of the class C(α), which has been widely studied in classical and biharmonic nonlinear Schrödinger equations. See [3,4,5,6,10,11,14,15,16,20,21,22] for instance.…”