$ H^2 $ blowup result for a Schrödinger equation with nonlinear source term

Abstract

In this paper, we consider the nonlinear Schrödinger equation on R N , N ≥ 1, ∂tu = i∆u + λ|u| α u, with H 2 -subcritical nonlinearities: α > 0, (N − 4)α < 4 and Reλ > 0. For any given compact set K ⊂ R N , we construct H 2 solutions that are defined on (−T, 0) for some T > 0, and blow up exactly on K at t = 0. We generalize the range of the power α in the result of Cazenave, Han and Martel [5]. The proof is based on the energy estimates and compactness arguments.

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

“…These formal calculations can be justified by standard approximation arguments, see e.g. Proposition 3.1 in [27]. Applying Cauchy-Schwartz inequality, (6.9), (6.16) and (7.7), we obtain…”

“…In fact, the proof is an obvious adaptation of [10,27]. More precisely, we just need to replace all 2 k with 4 k throughout the proof of Lemma 3.2 in [10] and Lemma 2.2 in [27] by considering the presence of ∆ 2 U j . Finally, note that 0 ≤ J ≤ k−6 4 by (6.2), we complete the proof of Lemma 6.2 by setting j = J. Lemma 6.3.…”

“…After that, by refining the initial ansatz U 0 inductively, the blow-up result is extended to the whole range of H 1 subcritical powers and arbitrary Imλ < 0 in Cazenave, Han and Martel [10]. This result is then extended into H 2 -subcritical case: α > 0 and (N − 4)α < 4 in [27] with the additional technical assumption −Imλ > α 2 |Reλ|. We prove Theorem 1.3 by applying the strategy of [12].…”

A Nonlinear Fractional Problem with Mixed Volterra-Fredholm Integro-Differential Equation: Existence, Uniqueness, H-U-R Stability, and Regularity of Solutions

“…Over the last few decades, there has been significant development in the area of ordinary and partial fractional differential equations with boundary integral conditions. This is reflected by various state-of-the-art papers, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Problems with initial integral conditions have many important applications.…”

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

“…(1.6), and then obtained a threshold value separating the existence and nonexistence of solutions. For the other types of second-order Schrödinger equation, many experts have paid attention to their qualitative properties and obtained abundant theoretical results, we refer the reader to see [30,19,2,36,32,7,6,1] and the papers cited therein.…”