Schrödinger-type identity for Schrödinger free boundary problems

This paper is concerned with the global existence of solutions for the semilinear nonlocal fractional Cauchy problem of the Schrödinger equation. Firstly, based on the Schrödinger approximation technique and the theory of a family of potential wells, the authors obtain the invariant sets and vacuum isolating of global solutions including the critical case. Then, the global existence of solutions and the stability of equilibrium points are discussed. Finally, the global asymptotic stability of the unique positive equilibrium point of the system is proved by applying the Leray-Schauder alternative fixed point theorem.

“…Using the Strichartz-type estimates and Gagliardo-Nirenberg's inequalities, Zhang et al [14] proved the existence of a global solution to the Cauchy problem…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we use a modified Schrödinger-type identity posted by Zhang et al [14] and prove the existence of global solutions for the semilinear nonlocal fractional Cauchy problem of the Schrödinger equation.…”

Section: Introductionmentioning

confidence: 99%

RETRACTED ARTICLE:Existence of global solutions for the semilinear nonlocal fractional Cauchy problem of the Schrödinger equation

Liu¹

2020

“…The regularity of the solution of the problem [24] with respect to the Schrödinger operator was discussed in [34] (see also [13,20,21,31,36]), where it was proved that ω ∈ C 0 ([0, L]; L p ‫))ג(‬ for all p ∈ [1, +∞) in the class of free boundary problems with respect to the Schrödinger operator of types 7) and that f ∈ C 0 ([0, M]; L p ‫))ג(‬ for all p ∈ [1,2] in the second-order class. More results as regards Schrödinger-type equations, wavelet analysis, distribution theory and calculus of variations were studied in previous work [16,18,19,27,32,50]. The semilinear elliptic equation on R n was considered in [7].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear conservation laws for the Schrödinger boundary value problems of second order

Ren

Yun

2020

In this paper, we apply a reliable combination of maximum modulus method with respect to the Schrödinger operator and Phragmén-Lindelöf method to investigate nonlinear conservation laws for the Schrödinger boundary value problems of second order. As an application, we prove the global existence to the solution for the Cauchy problem of the semilinear Schrödinger equation. The results reveal that this method is effective and simple.

“…In this paper we consider a singular boundary value problem with mixed boundary conditions and spatial heterogeneities given by (see [4][5][6]) f = χf in ‫,ג‬ f = 0 on 1 , ∂f + V (t)f = χω(t)u q on 2 , q > 1, (1) where:…”

Section: Introductionmentioning

confidence: 99%

“…The classical Randon transform (see [17]) is defined by the Cauchy principal value of the singular integral. In 2018, Zhang et al in [1] established the Schrödinger-type identity and applied it to a Schrödinger integral equation and then gave three examples where the kernel function is a Green's function for a two-order system of Schrödinger integral equations.…”

Section: Introductionmentioning

confidence: 99%

A modified Schrödinger-type identity: uniqueness of solutions for singular boundary value problem for the Schrödinger equation

Pang

2019

This paper is devoted to modifying the Schrödinger-type identity related to singular boundary value problem in (Zhang et al. in Bound. Value Probl. 2018:135, 2018). We also present some mathematical consequences of the method, including a stability result. The main technical tools used to develop the mathematical analysis are local and global bifurcation, monotonicity techniques, fixed point theory in b-metric spaces in (Liu et al. in Bull. Aust. Math. Soc. 94(1):121-130, 2016) and the maximum principle approach with respect to the Schrödinger operator in (Fan et al. in Math. Appl. 31(1):42-48, 2018). As an application, the uniqueness of solutions for singular boundary value problem for the Schrödinger equation is proved.