On Blowup Solutions to the Focusing Intercritical Nonlinear Fourth-Order Schrödinger Equation

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

“…The study of nonlinear fourth-order Schrödinger equation with the power type nonlinearity as in (1.1) has been attracted a lot of interest in the past decade, see e.g. [6,14,19,20,32,33,37] and references therein.…”

“…Recently, it has been proved rigorous in Boulenger [6], under the natural criteria from the well-known blowup results for the classical nonlinear Schrödinger equation, that the blowup solutions exist for the radial data in H 2 (R N ) with the negative initial energy. For more blowup results for the fourth-order Schrödinger equation, we refer to [2,3,4,14,20] and references therein.…”

“…20) where we used the boundedness of u n , u in (4.15) and(4.18). This inequality together with the smallness of T in (4.16) yields(4.19).Next, it follows from Lemma 3.1, (4.19) and the boundedness of u n , u again, we have, for any R > 0 lim sup…”

Random data theory for the cubic fourth-order nonlinear Schrödinger equation

“…In [5], Boulenger-Lenzmann established the existence of finite time blow-up H 2 -solutions for the focusing problem. Dynamical properties such as mass-concentration and limiting profile of blow-up H 2 -solutions were studied by Zhu-Yang-Zhang [37] and author [13].…”

On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term

“…(1.3) by establishing the profile decomposition of bounded sequences in H 2 (R n ). Dinh [8] established the profile decomposition of bounded sequences in Ḣγc ∩ Ḣ2 , and discussed the blow-up concentration and the limiting profile of blow-up solutions with critical Ḣγc -norm, where γ c = np−8 2p and 8 n < p < 8 (n−4) + ( 8 (n−4) + = +∞ when n ≤ 4 and 8 (n−4) + = 8 n−4 when n ≥ 5). In [22], Miao and Zhang consider the following higher-order nonlinear Schrödinger type equations…”