Article
Mathematics, Applied
Van Duong Dinh
Summary: This paper investigates the Cauchy problem for nonlinear Schrodinger equations as an effective model of the Bose-Einstein condensate in a magnetic trap rotating with an angular velocity. The study establishes sufficient conditions for the existence of global-in-time and finite time blow-up solutions, derives sharp thresholds for global existence versus finite time blow-up in different mass cases, and examines the existence and strong instability of ground state standing waves. Finally, the paper proves the existence, non-existence, and orbital stability of prescribed mass standing waves when the rotational speed is smaller than a critical value.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Article
Mathematics, Applied
Shuxin Ge, Rong Yuan, Xiaofeng Zhang
Summary: This paper studies an initial boundary value problem for a nonlocal parabolic equation with a diffusion term and convex-concave nonlinearities. By establishing the Lq-estimate and analyzing its energy, the existence of global solutions is proven and some blow-up conditions are obtained. Using the variational structure of the problem, the Mountain-pass theorem is utilized to demonstrate the existence of nontrivial steady-state solutions. The dynamical behavior of global solutions with relatively compact trajectories in H01 (Ω) is also established, showing uniform convergence to a non-zero steady state after a long time due to the energy functional satisfying the P.S. condition. Finally, an unstable steady states sequence is derived using another minimax theorem.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Bolys Sabitbek, Berikbol T. Torebek
Summary: In this study, we investigate a double nonlinear porous medium equation with a novel nonlinearity condition in a bounded domain. We introduce the blow-up solution for the problem under consideration with negative initial energy and construct invariant sets of solutions using a set of potential wells. We also analyze the global existence and asymptotic behavior of weak solutions, as well as the occurrence of blow-up phenomena within a finite time for the positive solution of the double nonlinear porous medium equation.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Applied
Deng Wang, Han Yang
Summary: The paper investigates the local and global existence of the inhomogeneous nonlinear Schrodinger equation with a specific nonlinearity function. It shows that a global solution exists in the mass-subcritical case for large data in the spaces L-p, p < 2, under certain conditions on the parameters. The solution is established using a data-decomposition argument, generalized Strichartz estimates in Lorentz spaces, and an interpolation theorem.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Nakao Hayashi, Chunhua Li, Takayoshi Ogawa, Takuya Sato
Summary: In this paper, we study the nonlinear initial boundary value problem of one dimensional Schrodinger equation with the power nonlinearity of order q in boundary conditions. We prove that if q > 2 and the initial data are small in suitable norms, then we have a global in time of solutions in a scale invariant space. Furthermore, we present a blow up result of small amplitude solutions when 1 < q <= 2 and derive an upper-bound of existence time. Our results indicate that the quadratic nonlinearity is a critical exponent.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2023)
Article
Mathematics, Applied
Jinmyong An, Roesong Jang, Jinmyong Kim
Summary: In this paper, the focusing inhomogeneous nonlinear Schrodinger equation with inverse-square potential is studied. The criteria for global existence and blow-up of solutions to the equation are established, and the global existence and blow-up of solutions with different data situations are investigated.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2023)
Article
Mathematics, Applied
Van Duong Dinh
Summary: We revisit the finite time blow-up problem for the fourth-order Schrodinger equation with a specific nonlinearity, proving the existence of non-radial blow-up solutions with negative energy using localized virial estimates and spatial decay of the nonlinearity. This result is the first one dealing with non-radial blow-up solutions to the fourth-order Schrodinger equations.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Yuan Shan, Baoqing Liu
Summary: This paper investigates the nonlinear Schrodinger-Kirchhoff equations on the whole space using variational methods, showing the existence and multiplicity of solutions for the problem with asymptotically linear nonlinearity through the Morse index of the reduced Schrodinger operator.
Article
Mathematics
Jichao Wang, Ting Yu
Summary: In this paper, we study the singular perturbation problem for the Schrodinger-Poisson equation with critical growth. We establish the relationship between the number of solutions and the profiles of the coefficients when the perturbed coefficient is small. Additionally, we observe a different concentration phenomenon without any restriction on the perturbed coefficient and obtain an existence result.
Article
Mathematics, Applied
Zhanwei Gou, Jincheng Shi
Summary: In this paper, the authors investigate a parabolic problem with nonlinear Neumann boundary conditions. They derive lower bounds for the blow-up time of the solution using a modified differential inequality in higher dimensional spaces. Under appropriate assumptions, they also provide an upper bound for the blow-up time.
Article
Physics, Mathematical
Riccardo Adami, Filippo Boni, Raffaele Carlone, Lorenzo Tentarelli
Summary: This article studies the ground state properties of the Schrodinger equation with a focusing nonlinearity and a point interaction. The research shows that ground states exist for every value of the mass and have certain characteristic properties such as positivity, radial symmetry, etc.
JOURNAL OF MATHEMATICAL PHYSICS
(2022)
Article
Mathematics
Aynur Bulut, Benjamin Dodson
Summary: Global well-posedness and scattering results for the logarithmically energy-supercritical nonlinear wave equation are established under the assumption that the initial data satisfies a partial symmetry condition. These results generalize and extend previous work by Tao in the radially symmetric setting, utilizing techniques involving weighted versions of Morawetz and Strichartz estimates with weights adapted to partial symmetry assumptions. Additionally, a corresponding quantitative result for the energy-critical problem is established in an appendix.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2021)
Article
Mathematics, Applied
Yue Pang, Vicentiu D. Radulescu, Run Zhang Xu
Summary: In this paper, we study the initial boundary value problem of m-Laplacian parabolic equation in various physical models. We analyze the problem for three different initial energy levels. For sub-critical energy, we find the blow-up result and estimate the lower and upper bounds of the blow-up time. For critical energy, we establish global existence, asymptotic behavior, finite time blow-up, and the lower bound of the blow-up time. For super-critical energy, we prove finite time blow-up and estimate the lower and upper bounds of the blow-up time.
ACTA MATHEMATICA SINICA-ENGLISH SERIES
(2023)
Article
Mathematics, Applied
Xiao Wei
Summary: In this paper, we study the global existence and blow up phenomenon of certain hyperbolic systems. We prove the global existence of solutions and exponential decay of solutions by using the method of modified potential well and introducing an appropriate Lyapunov function. Moreover, we discuss the blow-up behavior of weak solutions and provide estimates for the lifespan of solutions using the concave method.
ANALYSIS AND MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
Sehrish Javed, Mokhtar Kirane, Salman A. Malik
Summary: This article considers nonlinear integro-differential inequalities involving two arbitrary kernels. The results presented in this article generalize some previous cases. The non-existence of a global solution in time has been proven. The proof method is based on the weak formulation of the problem.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
