Article
Mathematics, Applied
Mykael Cardoso, Luiz Gustavo Farah, Carlos M. Guzman
Summary: The research focuses on the inhomogeneous nonlinear Schrodinger equation in R-N and establishes global results in H-1. Local well-posedness and global existence of solutions are studied for different dimensional cases and parameter ranges. The concentration phenomenon of L-sigma c-norm is also analyzed, with a method based on compact embedding into a weighted L2 sigma+2 space.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Naoki Matsui
Summary: This paper discusses the existence and behavior of blow-up solutions for the threshold of critical mass in a nonlinear Schrodinger equation with an inverse potential. A critical-mass finite-time blow-up solution is constructed, and it is shown that the blow-up solution converges to a certain blow-up profile in the virial space.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2021)
Article
Mathematics
Hideo Takaoka
Summary: The study focused on the Cauchy problem of the mass critical nonlinear Schrodinger equation with derivative and a mass of 4 pi. Global well-posedness was proven in H-1 under certain conditions, and the limiting profile of blow up solutions with the critical 4 pi mass was obtained using the concentration compact principle as originally done by Kenig-Merle.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Qing Guo, Hua Wang, Xuewen Wang
Summary: This study demonstrates that assuming certain initial data leads to blow-up solutions of the inter-critical defocusing nonlinear Schrodinger equation, with the existence of initial data of minimal -norm producing blow-up. Furthermore, the set of such data is compact in a certain space up to invariant transformations. The main methodology involves profile decomposition combined with perturbation argument.
APPLICABLE ANALYSIS
(2021)
Article
Mathematics, Applied
Jingjing Pan, Jian Zhang
Summary: This paper investigates the mass-critical variable coefficient nonlinear Schrodinger equation, and explores the existence, compactness, and uniqueness of the ground state solutions.
ADVANCES IN NONLINEAR ANALYSIS
(2022)
Article
Mathematics
Mykael Cardoso, Carlos M. Guzman, Ademir Pastor
Summary: In this article, we study the local well-posedness and sufficient conditions for global existence of solutions for the inhomogeneous biharmonic nonlinear Schr o dinger equation, and investigate the phenomenon of norm concentration for finite time blow up solutions. The main tool used in this study is the compact embedding of (L)over dot(p) boolean AND (H)over dot(2) into a weighted L2σ+2 space, which is of independent interest.
MONATSHEFTE FUR MATHEMATIK
(2022)
Article
Mathematics, Applied
Lu Tao, Yajuan Zhao, Yongsheng LI
Summary: In this paper, we investigate the well-posedness and blow-up solutions of the fractional Schrödinger equation with a Hartree-type nonlinearity and subcritical or critical perturbations. We prove the local well-posedness for both defocusing and focusing cases with subcritical or critical nonlinearity, for nonradial or radial initial data. We obtain the global well-posedness for the defocusing case, and for the focusing mass-subcritical or mass-critical case with small enough initial data. We also study the blow-up solutions for the mass-critical problem.
JOURNAL OF PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
Yiming Su, Deng Zhang
Summary: We study the focusing mass-critical rough nonlinear Schrodinger equations with controlled rough path stochastic integration. We construct minimal mass blow-up solutions in dimensions one and two, which behave similarly to pseudo-conformal blow-up solutions near the blow-up time. Furthermore, we establish global well-posedness for initial data with mass below the ground state. These results show that the mass of the ground state is the exact threshold for global well-posedness and blow-up in the stochastic focusing mass-critical case. Similar results are also obtained for a class of nonlinear Schrodinger equations with lower order perturbations.
JOURNAL OF FUNCTIONAL ANALYSIS
(2023)
Article
Mathematics, Applied
Vo Van Au, Yong Zhou, Donal O'Regan
Summary: This paper considers the Cauchy problem for a semilinear biparabolic equation and discusses the existence and properties of solutions under different Lipschitz conditions.
MEDITERRANEAN JOURNAL OF MATHEMATICS
(2022)
Article
Mathematics, Applied
Tarek Saanouni, Congming Peng
Summary: This note investigates the local existence and well-posedness of the inhomogeneous Schrödinger equation with different degrees of homogeneity. The analysis involves Sobolev critical exponents and weighted Lebesgue spaces. This work extends the existing literature on this topic.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Yan Rybalko, Dmitry Shepelsky
Summary: In this paper, the Cauchy problem for the integrable nonlocal nonlinear Schrödinger equation is considered. The main aim is to propose a suitable concept for the continuation of solutions, including possible singularities, using the inverse scattering transform method.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Applied
Marcelo M. Cavalcanti, Valeria N. Domingos Cavalcanti
Summary: This paper investigates the existence and uniform decay rates of the energy associated with the nonlinear damped Schrodinger equation, using a different approach than previous studies by focusing on nonlinear damping instead of pseudo-differential operator properties. No growth assumptions on g(z) near the origin are made in the analysis.
ADVANCED NONLINEAR STUDIES
(2021)
Article
Mathematics, Applied
Anudeep K. Arora, Oscar Riano, Svetlana Roudenko
Summary: We investigate the well-posedness of the generalized Hartree equation under low powers of nonlinearity. By using weighted Sobolev spaces and the boundedness of the Riesz transform, we establish the local and global existence of a class of data, with scattering in positive time. Furthermore, in the focusing case in the L-2-supercritical setting, we prove the blow-up of a subset of locally well-posed data with positive energy in finite time.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2022)
Article
Mathematics, Applied
Tatsuya Hosono, Takayoshi Ogawa
Summary: This paper considers the Cauchy problem for an attraction-repulsion chemotaxis system with chemotactic coefficients of the attractant beta(1) and the repellent beta(2) in Rn. The coefficients play an important role in the global existence and blow up of solutions. The paper demonstrates the local well-posedness of solutions in the critical spaces L-n/2(R-n) and the finite time blow-up of the solution under the condition beta(1) > beta(2) in higher dimensional spaces.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics
Nobu Kishimoto
Summary: The study examines the uniqueness of solutions and specific scale-subcritical regularities for nonlinear Schrödinger equations with nonlinear terms on the d-dimensional torus. The proof relies on various multilinear estimates and the infinite normal form reduction argument.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
