On the Dirac–Klein–Gordon Equations in Three Space Dimensions

ABSTRACT

We establish a local existence result for Dirac–Klein–Gordon equations in three space dimensions, employing a null form estimate and a fixed point argument.     

Global attractors for the discrete Klein–Gordon–Schrödinger type equations

The purpose of this work is to investigate the asymptotic behaviours of solutions for the discrete Klein–Gordon–Schrödinger type equations in one-dimensional lattice. We first establish the global existence and uniqueness of solutions for the corresponding Cauchy problem. According to the solution's estimate, it is shown that the semi-group generated by the solution is continuous and possesses an absorbing set. Using truncation technique, we show that there exists a global attractor for the semi-group. Finally, we extend the criteria of Zhou et al. [S. Zhou, C. Zhao, and Y. Wang, Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems, Discrete Contin. Dyn. Syst. A 21 (2008), pp. 1259–1277.] for finite fractal dimension of a family of compact subsets in a Hilbert space to obtain an upper bound of fractal dimension for the global attractor.     

The Radial Defocusing Nonlinear Schrödinger Equation in Three Space Dimensions

We study the defocusing nonlinear Schrödinger equation in three space dimensions. We prove that any radial solution that remains bounded in the critical Sobolev space must be global and scatter. In the energy-supercritical setting we employ a space-localized Lin–Strauss Morawetz inequality of Bourgain. In the intercritical regime we prove long-time Strichartz estimates and frequency-localized Lin–Strauss Morawetz inequalities.     

Dispersive estimates for 1D discrete Schrödinger and Klein–Gordon equations

We derive the long-time asymptotics for solutions of the discrete 1D Schrödinger and Klein–Gordon equations.     

Low regularity for the nonlinear Klein–Gordon systems

In this paper, we study the global well-posedness below the energy norm of the Cauchy problem for the Klein–Gordon system in R³${\mathbb{R}}^3$. We prove the H^s$H^s$-global well-posedness with s<1$s<1$ of the Cauchy problem for the Klein–Gordon system. The method invoked is different from the well-known Bourgain’s method [Jean Bourgain, Refinements of Strichartz’s inequality and applications to 2D-NLS with critical nonlinearity, International Mathematial Research Notices 5 (1998) 253–283].     

A linear,symmetric and energy-conservative scheme for the space-fractional Klein–Gordon–Schrödinger equations

In this paper, we propose an efficient numerical scheme for the space-fractional Klein–Gordon–Schrödinger (SFKGS) equations. Motivated by the “Invariant Energy Quadratization” (IEQ) approach, we introduce two auxiliary variables to transform the SFKGS system into a new equivalent system in which the time derivative is discretized by the Crank–Nicolson method, and the space discretization is based on the Fourier spectral method. Consequently, the numerical scheme shares two good features. The first feature is that the nonlinear terms are treated semi-explicitly and a linear symmetric system is solved at each time step. The second feature is the energy conservation at the discrete level. These two advantages are proved by the theoretical analysis and illustrated by a given numerical example.     

The Cauchy Problem for the Fractional Diffusion Equation in a Weighted Hölder Space

Under study is the Cauchy problem for the fractional diffusion equation with a Caputo derivative. The existence and uniqueness theorems for a smooth solution are proven in a weighted H¨older space.     

The Cauchy problem for non-autonomous nonlinear Schrödinger equations

In this paper we study the Cauchy problem for cubic nonlinear Schrödinger equation with space- and time-dependent coefficients on ∝^m and \(\mathbb{T}^m \). By an approximation argument we prove that for suitable initial values, the Cauchy problem admits unique local solutions. Global existence is discussed in the cases of m = 1, 2.     

High-order linear compact conservative method for the nonlinear Schrödinger equation coupled with the nonlinear Klein–Gordon equation

In this paper, we design a linear-compact conservative numerical scheme which preserves the original conservative properties to solve the Klein–Gordon–Schrödinger equation. The proposed scheme is based on using the finite difference method. The scheme is three-level and linear-implicit. Priori estimate and the convergence of the finite difference approximate solutions are discussed by the discrete energy method. Numerical results demonstrate that the present scheme is conservative, efficient and of high accuracy.     

Blow-up of Solutions of the Cauchy Problem for a Nonlinear Schrödinger Evolution Equation

The solution of the Cauchy problem for a nonlinear Schrödinger evolution equation with certain initial data is proved to blow up in a finite time, which is estimated from above. Additionally, lower bounds for the blow-up rate are obtained in some norms.     

The Cauchy Problem for Defocusing Nonlinear Schrödinger Equations with Non-Vanishing Initial Data at Infinity

For rather general nonlinearities, we prove that defocusing nonlinear Schrödinger equations in ? ⁿ (n ≤ 4), with non-vanishing initial data at infinity u ₀, are globally well-posed in u ₀ + H ¹. The same result holds in an exterior domain in ? ⁿ , n = 2, 3.     

Spectral method for solving the system of equations of Schrödinger—Klein—Gordon field

In this paper, we construct the conservative spectral scheme for the periodic initial-value problem for a system of equations of the complex Schrödinger field, interacting with the real Klein—Gordon field and estimate the error which is \xV;\GF;(nk) − Φ^n_N\xV; + \xV;χ(nk) − iXⁿ_n\xV;₁ = O(k² + N^−(γ−1)).     

Multiple solutions for nonhomogeneous Schrödinger–Maxwell and Klein– Gordon–Maxwell equations on R 3

In this paper we study the following nonhomogeneous Schrödinger–Maxwell equations $\left\{\begin{array}{ll} {-\triangle u+V(x)u+ \phi u=f(x,u)+h(x),} \quad {\rm in}\,\,\,{\mathbf{R}}^3,\\ {-\triangle \phi=u^2, \qquad\qquad\qquad\qquad\qquad\qquad\,\,\, {\rm in} \,\,{\mathbf{R}}^3,} \end{array} \right.$ where f satisfies the Ambrosetti–Rabinowitz type condition. Under appropriate assumptions on V, f and h, the existence of multiple solutions is proved by using the Ekeland’s variational principle and the Mountain Pass Theorem in critical point theory. Similar results for the nonhomogeneous Klein–Gordon–Maxwell equations $\left\{\begin{array}{ll} {-\triangle u+[m^2-(\omega+\phi)^2]u=|u|^{q-2}u+h(x), \quad {\rm in} \,\,\,{\mathbf{R}}^3,}\\ {-\triangle \phi+ \phi u^2=-\omega u^2, \qquad\qquad\qquad\qquad\qquad\,\,\, {\rm in} \,\,\,{\mathbf{R}}^3,} \end{array} \right.$ are also obtained when 2 < q < 6.     

Energy decay for the linear Klein–Gordon equation and boundary control

In this work we study the asymptotic behavior of the solutions of the linear Klein–Gordon equation in R^N${\mathbb{R}}^N$, N?1$N?1$. We prove that local energy of solutions to the Cauchy problem decays polynomially. Afterwards, we use the local decay of energy to study exact boundary controllability for the linear Klein–Gordon equation in general bounded domains of R^N${\mathbb{R}}^N$, N?1$N?1$.