On well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schrödinger equation

Abstract

We prove the local well-posedness for the nonlinear fourth-order Schrödinger equation (NL4S) in Sobolev spaces. We also studied the regularity of solutions in the sub-critical case. A direct consequence of this regularity is the global well-posedness above mass and energy spaces under some assumptions. Finally, we show the ill-posedness for (NL4S) in some cases of the super-critical range.

Bilinear Strichartz's type estimates in Besov spaces with application to inhomogeneous nonlinear biharmonic Schrödinger equation

“…Remark 1.8. Corollary 1.7 improves the corresponding results of [7,9] in the case 0 < s ≤ 2. In [7], there is a additional assumption s ≤ [α] ([α] denotes the largest integer less than or equal to α) for the parameter α; and in [8], the continuous dependence (u n → u in C([0, T ], H s−ε ) with ε > 0) is weaker than the expected one.…”

Global well-posedness for a L2-critical nonlinear higher-order Schrödinger equation

“…The conservations of mass and energy combine with the persistence of regularity (see e.g. [11]) immediately yield the global well-posedness for (NLS k ) with initial data in H γ (R k ) when γ ≥ k/2. Note also (see [10]) that one has the local well-posedness for (NLS k ) when initial data u 0 ∈ L 2 (R k ) but the time of existence depends not only on the size but also on the profile of the initial data.…”

Regularity properties of the cubic biharmonic Schrödinger equation on the half line

“…For the other well-posedness results related to Eq. (3) see for instance [1,10,[20][21][22][23][24]. Initial boundary value problems for the fourth order NLS have been recently started to be addressed.…”