Integral transform approach to solving Klein–Gordon equation with variable coefficients

In this paper we describe the integral transform that allows to write solutions of the time‐dependent partial differential equation via solution of a simpler equation. This transform was suggested by the author in the case when the last equation is a wave equation, and then it was used to investigate several well‐known equations such as Tricomi‐type equation, the Klein–Gordon equation in the de Sitter and Einstein‐de Sitter spacetimes. A generalization given in this paper allows us to consider also the Klein–Gordon equations with coefficients depending on the spatial variables.     

对流反应扩散方程的稳定化时间间断时空有限元解的误差估计

在流线迎风Petrov-Galerkin（SUPG）稳定化有限元数值格式的基础上,结合时间方向的变分离散,构造对流反应扩散方程的稳定化时间间断时空有限元格式.该类格式在工程上有一些数值模拟应用,但相关文献没有看到类似数值格式的理论证明.本文以Radau点为节点,构造时间方向的Lagrange插值多项式,证明了稳定化有限元解的稳定性,时间最大模、空间L²（Ω）-模误差估计.文中利用插值多项式和有限元方法相结合的技巧,解耦时空变量,去掉了时空网格的限制条件,提供了时间间断稳定化时空有限元方法的理论证明思路,克服了因时空变量统一导致的实际计算时的复杂性.     

Global bifurcation of solutions of the mean curvature spacelike equation in certain Friedmann–Lemaître–Robertson–Walker spacetimes

We study the existence of spacelike graphs for the prescribed mean curvature equation in the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. By using a conformal change of variable, this problem is translated into an equivalent problem in the Lorentz–Minkowski spacetime. Then, by using Rabinowitz's global bifurcation method, we obtain the existence and multiplicity of positive solutions for this equation with 0-Dirichlet boundary condition on a ball. Moreover, the global structure of the positive solution set is studied.     

On the Ricci Curvature of Compact Spacelike Hypersurfaces in Einstein Conformally Stationary-Closed Spacetimes

In this paper we develop an integral formula involving the Ricci and scalar curvatures of a compact spacelike hypersurface M in a spacetime equipped with a timelike closed conformal vector field K (in short, conformally stationary-closed spacetime), and we apply it, when is Einstein, in order to establish sufficient conditions for M to be a leaf of the foliation determined by K and to obtain some non-existence results. We also get some interesting consequences for the particular case when is a generalized Robertson-Walker spacetime.     

Semigroup Properties and the Crandall Liggett Approximation for a Class of Differential Equations with State-Dependent Delays

We present an approach for the resolution of a class of differential equations with state-dependent delays by the theory of strongly continuous nonlinear semigroups. We show that this class determines a strongly continuous semigroup in a closed subset of C^0, 1. We characterize the infinitesimal generator of this semigroup through its domain. Finally, an approximation of the Crandall-Liggett type for the semigroup is obtained in a dense subset of (C, ‖·‖_∞). As far as we know this approach is new in the context of state-dependent delay equations while it is classical in the case of constant delay differential equations.     

CAUSAL DOMAIN OF MINKOWSKI-SPACETIME TANGENT BUNDLE

The geometry of the causal domain of the spacetime tangent bundle is examined for Finslerian quantum fields in Minkowski spacetime. The Planck-scale structure of the boundary of the causal domain is elaborated. The geometry indicates that at the Planck scale, causal connectivity of Finslerian quantum fields may occur between spacelike separated points, and also at larger scales for extremely large relative four-velocities.     

An Exact Interior Solution of a Static Charged Sphere in Higher Dimensional Spacetime

By assuming "p = Aρ" and the total intrinsic energy density εr^k (k + D - 1 > 0), we strictly solve the higher dimensional Einstein-Maxwell field equations in a static charged sphere, resulting in an exact interior solution of the equations.     

Singularity structure of the two point function of the free Dirac field on a globally hyperbolic spacetime

We give an introduction to the techniques from microlocal analysis that have successfully been applied in the investigation of Hadamard states of free quantum field theories on curved spacetimes. The calculation of the wave front set of the two point function of the free Klein‐Gordon field in a Hadamard state is reviewed, and the polarization set of a Hadamard two point function of the free Dirac field on a curved spacetime is calculated.     

Exact Solutions of Dirac Equation and Particle Creation in (1+3)-Dimensional Robertson-Walker Spacetime

The exact solutions of the Dirac equation are discussed for a Robertson-Walker spacetime with asymptotically Minkowskian in and out regions. We obtain the mode solutions which reduce to positive and negative Minkowskian spinors in asymptotically regions. Using the obtained solutions we compute the density of created particles.     

Particle production in curved spacetime

Particle production in curved spacetime has been discussed through the method of complex time WKB approximation. We consider Dirac equation in non-flat spacetime to understand particle production as particle-antiparticle rotation. The method is also generalized to understand particle production through parametric resonance. To understand the method of CWKB we consider particle production in Kasner spacetime as an example.