On Energy Critical Inhomogeneous Bi-Harmonic Nonlinear Schrödinger Equation
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In this note, one focus is on the inter-critical regime \ (0 Abstract. In this work, we consider the three-dimensional defocusing energy-critical nonlinear Schrödinger equation . Applying the incoming and outgoing decomposition presented in the recent work [M. Beceanu, Q. Deng, A. Soffer, and Y. Wu, Comm. Math. Phys., 382 (2021), pp. 173–237], we prove that for any radial function with and with , there exists an outgoing This book is suitable as the basis for a one-semester course, and serves as a useful introduction to nonlinear Schrödinger equations for those with a background in harmonic analysis, functional analysis, and partial differential equations. This work proves the local existence of solutions to some inhomogeneous nonlinear equations of Schrödinger type with a fractional Laplacian in Sobolev spaces. Moreover, for small datum, the local solution extends to a global one. We give an elementary proof based on Strichartz estimates coupled with a fix point argument. In order to avoid a loss of regularity in Strichartz estimates, Stability of standing waves for nonlinear Schrödinger equations with potentials Séminaire É. D. P. (2003-2004), Exposé no IX, 8 p. We prove pointwise-in-time dispersive decay for solutions to the energy-critical nonlinear Schrödinger equation in spatial dimensions d=3,4for both the initial-value and final-state problems. In this paper, we study the Cauchy problem of the inhomogeneous nonlinear Schrödinger equation with a harmonic potential: i ∂ t u = div (f (x) ∇ u) + | x | 2 u k (x) | u | 4 / N u , ∊ x ∊ R N , N ≥ 1, which models the remarkable Bose–Einstein condensation. We discuss the existence and nonexistence results and investigate the limiting profile of blow-up solutions with 1.3 Inhomogeneous Nonlinear Schrödinger Equations (Gross–Pitaevskii Equations) The discoveries of the past decades in both Bose–Einstein condensates (BECs) and in optical fiber, as well as in many other branches of physical sciences have involved much interest in the study of nonlinear partial differential equations of Schrödinger type. 2(4−τ) solutions for 1 < p < 1 + with small data. These results are known to be N−4 false for the gauge invariant problem (1.3). To the author knowledge, this work is the first one dealing with the inhomogeneous nonlinear bi-harmonic Schr ̈odinger equation with non-gauge invariant source term (1.1). The plan of this note is as follows. Section 2 contains the main results and some We study long time dynamics of nonradial solutions to the focusing inhomogeneous nonlinear Schrödinger equation. By using the concentration/compactness and rigidity method, we establish a scatterin We investigate the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation i∂tu + Δ u = −| x | b | u | p −1u in the radial Sobolev space H r 1 (R N) , where b > 0 and p > 1. We show the global existence and energy scattering in the intercritical regime, i.e., p> N + 4 + 2 b N and p Request PDF | Energy scattering for radial focusing inhomogeneous bi-harmonic Schrödinger equations | This note studies the asymptotic behavior of global solutions to the fourth-order Abstract. In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing nonlinear Schrödinger equation with various (deterministic and random) initial data. We show that nonlinear solutions enjoy the same decay rate as the linear ones. The regularity assumption on the initial data is much lower than in previous results (see [C. Fan and In this paper, we study the Cauchy problem of the inhomogeneous energy-critical Schrödinger equation: i ∂ t u = – Δ u – k (x) | u | 4 N – 2 u, N ≥ 3. Using the potential well method, we establish some new sharp criteria for blow-up of solutions in the nonradial case. In particular, our conclusion in some sense improves on the results in [Kenig and Merle, Invent. Math. 166, We consider the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS) iut+Δ2u+λ|x|−b|u|αu=0, where λ=±1 and α, b>0. We show local and global well-posedness in Hs (RN) in the Hs Local well-posedness for the inhomogeneous nonlinear Schrödinger equation Article Jan 2021 DISCRETE CONT DYN S Lassaad Aloui Slim Tayachi Download Citation | Energy Scattering for Non-radial Inhomogeneous Fourth-Order Schrödinger Equations | It is the goal of this manuscript to establish the scattering of global solutions to the The challenge of this work is to extend [17], where the author establishes the scattering of defocusing global solutions to a class of inhomogeneous bi-harmonic Schrödinger equations. quantum field equations or black hole solutions of the Einstein’s equations [2 ]. For s = 2, the above equations are called fourth-order Schr ̈odinger equations. The bi-harmonic Schr ̈odinger problem was considered first in [22, 23] to take into account the role of small fourth-order dispersion terms i the propagation of intense We consider the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS) i u t + Δ 2 u + λ | x | − b | u | α u = 0, where λ = ± 1 and α, b> 0. We show local and global well-posedness in H s (R N) in the H s -subcritical case, with s = 0, 2. Moreover, we prove a stability result in H 2 (R N), in the mass-supercritical and energy-subcritical case. The The nonlinear Schrödinger equation is an equation that can be interpreted as the differential law of the development in time of a system. It is a very versatile model applicable to many disciplines in engineering and applied science, such as dynamical systems, nonlinear optics, fluid dynamics, materials science, statistical physics, particle physics, astrophysics, Hence, the scale-invariant Sobolev space is , with the critical index . If (equivalently ), the Cauchy problem (1.1) is known as -critical; if in particular (equivalently ), it is called energy-critical or -critical. The nonlinear biharmonic Schrödinger equation (1.1), also called the fourth-order Schrödinger equation, was introduced by Karpman [16], and In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schrödinger equation with spatial inhomogeneity coefficient K( View a PDF of the paper titled The 3D energy-critical inhomogeneous nonlinear Schrodinger equation with strong singularity, by Yoonjung Lee We prove scattering below the mass–energy threshold for the focusing inhomogeneous nonlinear Schrödinger equation i u t + Δ u + | x | b | u | p 1 u = 0, when b ≥ 0 and N> 2 in the intercritical case 0 In this paper we study dynamical properties of blowup solutions to the focusing inter-critical (mass-supercritical and energy-subcritical) nonlinear fourth-order Schrödinger equation. The equation in (1.1) is an inhomogeneous version of the well known biharmonic nonlinear Schrödinger equation (BNLS) (also termed fourth-order Schrödinger equation), see [1]. Moreover, the equation is called “focusing IBNLS” when λ = 1 and “defocusing IBNLS” when λ = 1. Key words and phr ases. Inhomogeneous fourth-order nonlinear Schr¨odinger equation,L2−critical, blow- up, localized virial id entity, non-radial solutions. In this paper, we study the critical norm problem for the defocusing inhomogeneous nonlinear Schrödinger equations (INLS). As a first attempt, we consider the defocusing -critical (INLS) with and in dimensions . We utilize the concentration-compactness/rigidity method as in Kenig-Merle [22] and Murphy [32] to show that if a solution u is uniformly bounded in the critical Conjecture 1.5. For arbitrary initial data u0 2 _H1 x(Rd), the defocusing energy-critical nonlinear Schrodinger equation is globally wellposed and solutions obey global spacetime bounds; in particular, scattering holds. Energy scattering for a class of inhomogeneous biharmonic nonlinear Schrödinger equations in low dimensions. 2023. hal-03926557 arXiv:2211.11824v2 [math.AP] 24 Nov 2022 ENERGY SCATTERING FOR A CLASS OF INHOMOGENEOUS BIHARMONIC NONLINEAR SCHRODINGER EQUATIONS IN LOW¨ DIMENSIONS VAN DUONG DINH AND SAHBI Abstract This work investigates the Cauchy problem for a focusing inhomogeneous nonlinear Hartree equation in the energy space \ (H^1_ {rd} (\mathbb R^N)\). The goal is three folds. First, one gives a sharp Gagliardo-Nirenberg type inequality adapted to the evolution problem. Second, one proves a local well-posedness result in the Pausader, B.: Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case. Dyn. Partial Differ. Equ. 4 (3), 197–225 (2007) The energy scattering implies that the energy global solutions to the considered equation are asymptotic to $ e^ {i\cdot\Delta^2}u_\pm $, when $ t\to\pm\infty $. Semantic Scholar extracted view of „A note on the mass-critical inhomogeneous generalized Hartree equation“ by S. Almuthaybiri et al. This work develops a local theory of the inhomogeneous coupled Schrödinger equations [Formula: see text]. Here, one treats the critical Sobolev regime [Formula: see text], where [Formula: seeElectronic Journal of Differential Equations, Vol. 2024 , No.
