In this article we discuss the maximum principle for the linear equation and the sign changing so... more In this article we discuss the maximum principle for the linear equation and the sign changing solutions of the semilinear equation with the Higgs potential. Numerical simulations indicate that the bubbles for the semilinear Klein-Gordon equation in the de Sitter space-time are created and apparently exist for all times.
In this article we discuss the maximum principle for the linear equation and the sign changing so... more In this article we discuss the maximum principle for the linear equation and the sign changing solutions of the semilinear equation with the Higgs potential. Numerical simulations indicate that the bubbles for the semilinear Klein-Gordon equation in the de Sitter space-time are created and apparently exist for all times.
Advanced studies in pure mathematics, Dec 29, 2020
This is a survey of a novel integral transform and its applications to solving second-order hyper... more This is a survey of a novel integral transform and its applications to solving second-order hyperbolic equations in the curved space-times.
We prove the existence of global in time solution with the small initial data for the semilinear ... more We prove the existence of global in time solution with the small initial data for the semilinear equation of the spin-1 2 particles in the Friedmann-Lemaître-Robertson-Walker spacetime. Moreover, we also prove that if the initial function satisfies the Lochak-Majorana condition, then the global solution exists for arbitrary large initial value. The solution scatters to free solution for large time. The mass term is assumed to be complex-valued. The conditions on the imaginary part of mass are discussed by proving nonexistence of the global solutions if certain relation between scale function and the mass are fulfilled.
In this paper, we prove the existence of global in time small data solutions of semilinear Klein-... more In this paper, we prove the existence of global in time small data solutions of semilinear Klein-Gordon equations in the space-time with a static Schwarzschild radius in the expanding universe. 1 The black hole in expanding universe. The model with static Schwarzschild radius. Main results
We consider waves, which obey the semilinear Klein-Gordon equation, propagating in the Friedmann-... more We consider waves, which obey the semilinear Klein-Gordon equation, propagating in the Friedmann-Lemaître-Robertson-Walker spacetimes. The equations in the de Sitter and Einstein-de Sitter spacetimes are the important particular cases. We show the global in time existence in the energy class of solutions of the Cauchy problem.
This is a survey of the author's recent work rather than a broad survey of the literature. The su... more This is a survey of the author's recent work rather than a broad survey of the literature. The survey is concerned with the global in time solutions of the Cauchy problem for matter waves propagating in the curved spacetimes, which can be, in particular, modeled by cosmological models. We examine the global in time solutions of some class of semililear hyperbolic equations, such as the Klein-Gordon equation, which includes the Higgs boson equation in the Minkowski spacetime, de Sitter spacetime, and Einstein & de Sitter spacetime. The crucial tool for the obtaining those results is a new approach suggested by the author based on the integral transform with the kernel containing the hypergeometric function.
This work is concerned with the proof of stability of global Gevrey solution to the following qua... more This work is concerned with the proof of stability of global Gevrey solution to the following quasilinear weakly hyperbolic equation: utt - a(x, t)uxx = f(x, t, u, ux ) in P × [0, T] wit h initial data u(x, 0) = u0(x) and ut(x, 0) = u1 (x). Here weak hyperbolicity means that a(x,t) >_ 0, that is, there exist, in general, characteristic roots of variable multiplicity. One has to distinguish between the case of spatial degeneracy and that of time degeneracy. The connection to the life span of solutions is given.
We prove the existence of a global in time solution of the semilinear Klein-Gordon equation in th... more We prove the existence of a global in time solution of the semilinear Klein-Gordon equation in the de Sitter space-time. 0 Introduction In this paper we prove the existence of a global in time solution of the semilinear Klein-Gordon equation in the de Sitter space-time. The metric g in the de Sitter space-time is defined as follows, g 00 = g 00 = −1, g 0j = g 0j = 0, g ij (x, t) = e 2t σ ij (x), i, j = 1, 2,. .. , n, where n j=1 σ ij (x)σ jk (x) = δ ik , and δ ij is Kronecker's delta. The metric σ ij (x) describes the time slices. In the quantum field theory the matter fields are described by a function ψ that must satisfy equations of motion. In the case of a massive scalar field, the equation of motion is the semilinear Klein-Gordon equation generated by the metric g: g ψ = m 2 ψ + V ′ ψ (x, ψ) .
High performance computations are presented for the Higgs Boson Equation in the de Sitter Spaceti... more High performance computations are presented for the Higgs Boson Equation in the de Sitter Spacetime using explicit fourth order Runge-Kutta scheme on the temporal discretization and fourth order finite difference discretization in space. In addition to the fully three space dimensional equation its one space dimensional radial solutions are also examined. The numerical code for the three space dimensional equation has been programmed in CUDA Fortran and was performed on NVIDIA Tesla K40c GPU Accelerator. The radial form of the equation was simulated in MATLAB. The numerical results demonstrate the existing theoretical result that under certain conditions bubbles form in the scalar field. We also demonstrate the known blow-up phenomena for the solutions of the semilinear Klein-Gordon equation with imaginary mass. Our numerical studies suggest several previously not known properties of the solution for which theoretical proofs do not exist yet: 1. smooth solution exists for all time if the initial conditions are compactly supported and smooth; 2. under some conditions no bubbles form; 3. solutions converge to step functions related to unforced, damped Duffing equations.
International journal of applied mathematics and statistics, 2008
We investigate the issue of the Lp − Lq estimates for the Gellerstedt operator and generalized Tr... more We investigate the issue of the Lp − Lq estimates for the Gellerstedt operator and generalized Tricomi operator in the domain where the operators are hyperbolic. A special attention is given to the case of the Cauchy problem with the initial data given outside of the set where characteristics coincide.
Nonlinear Differential Equations and Applications NoDEA
We prove the existence of global in time solution with the small initial data for the semilinear ... more We prove the existence of global in time solution with the small initial data for the semilinear equation of the spin-1 2 particles in the Friedmann-Lemaître-Robertson-Walker spacetime. Moreover, we also prove that if the initial function satisfies the Lochak-Majorana condition, then the global solution exists for arbitrary large initial value. The solution scatters to free solution for large time. The mass term is assumed to be complex-valued. The conditions on the imaginary part of mass are discussed by proving nonexistence of the global solutions if certain relation between scale function and the mass are fulfilled.
Springer Proceedings in Mathematics & Statistics, 2016
In this review, we present an integral transform that maps solutions of some class of the partial... more In this review, we present an integral transform that maps solutions of some class of the partial differential equations with time independent coefficients to solutions of more complicated equations, which have time-dependent coefficients. We illustrate this transform by applications to several model equations. In particular, we give applications to the generalized Tricomi equation, the Klein–Gordon and wave equations in the curved spacetimes such as Einstein-de Sitter, de Sitter, anti-de Sitter, and the spacetimes of the black hole embedded into de Sitter universe.
In this article we study global in time (not necessarily small) solutions of the equation for the... more In this article we study global in time (not necessarily small) solutions of the equation for the Higgs boson in the Minkowski and in the de Sitter spacetimes. We reveal some qualitative behavior of the global solutions. In particular, we formulate sufficient conditions for the existence of the zeros of global solutions in the interior of their supports, and, consequently, for the creation of the so-called bubbles, which have been studied in particle physics and inflationary cosmology. We also give some sufficient conditions for the global solution to be an oscillatory in time solution.
In this article we give sufficient and necessary conditions for the validity of the Huygens’ prin... more In this article we give sufficient and necessary conditions for the validity of the Huygens’ principle for the Dirac operator in the non-constant curvature spacetime of the Friedmann-Lemâıtre-RobertsonWalker models of cosmology. The Huygens’ principle discussed for the equation of a particle with mass m = 0 as well as a massive spin1 2 particle (field) undergoing a red shift of its wavelength as the universe expands.
In this article we discuss the maximum principle for the linear equation and the sign changing so... more In this article we discuss the maximum principle for the linear equation and the sign changing solutions of the semilinear equation with the Higgs potential. Numerical simulations indicate that the bubbles for the semilinear Klein-Gordon equation in the de Sitter space-time are created and apparently exist for all times.
In this article we discuss the maximum principle for the linear equation and the sign changing so... more In this article we discuss the maximum principle for the linear equation and the sign changing solutions of the semilinear equation with the Higgs potential. Numerical simulations indicate that the bubbles for the semilinear Klein-Gordon equation in the de Sitter space-time are created and apparently exist for all times.
Advanced studies in pure mathematics, Dec 29, 2020
This is a survey of a novel integral transform and its applications to solving second-order hyper... more This is a survey of a novel integral transform and its applications to solving second-order hyperbolic equations in the curved space-times.
We prove the existence of global in time solution with the small initial data for the semilinear ... more We prove the existence of global in time solution with the small initial data for the semilinear equation of the spin-1 2 particles in the Friedmann-Lemaître-Robertson-Walker spacetime. Moreover, we also prove that if the initial function satisfies the Lochak-Majorana condition, then the global solution exists for arbitrary large initial value. The solution scatters to free solution for large time. The mass term is assumed to be complex-valued. The conditions on the imaginary part of mass are discussed by proving nonexistence of the global solutions if certain relation between scale function and the mass are fulfilled.
In this paper, we prove the existence of global in time small data solutions of semilinear Klein-... more In this paper, we prove the existence of global in time small data solutions of semilinear Klein-Gordon equations in the space-time with a static Schwarzschild radius in the expanding universe. 1 The black hole in expanding universe. The model with static Schwarzschild radius. Main results
We consider waves, which obey the semilinear Klein-Gordon equation, propagating in the Friedmann-... more We consider waves, which obey the semilinear Klein-Gordon equation, propagating in the Friedmann-Lemaître-Robertson-Walker spacetimes. The equations in the de Sitter and Einstein-de Sitter spacetimes are the important particular cases. We show the global in time existence in the energy class of solutions of the Cauchy problem.
This is a survey of the author's recent work rather than a broad survey of the literature. The su... more This is a survey of the author's recent work rather than a broad survey of the literature. The survey is concerned with the global in time solutions of the Cauchy problem for matter waves propagating in the curved spacetimes, which can be, in particular, modeled by cosmological models. We examine the global in time solutions of some class of semililear hyperbolic equations, such as the Klein-Gordon equation, which includes the Higgs boson equation in the Minkowski spacetime, de Sitter spacetime, and Einstein & de Sitter spacetime. The crucial tool for the obtaining those results is a new approach suggested by the author based on the integral transform with the kernel containing the hypergeometric function.
This work is concerned with the proof of stability of global Gevrey solution to the following qua... more This work is concerned with the proof of stability of global Gevrey solution to the following quasilinear weakly hyperbolic equation: utt - a(x, t)uxx = f(x, t, u, ux ) in P × [0, T] wit h initial data u(x, 0) = u0(x) and ut(x, 0) = u1 (x). Here weak hyperbolicity means that a(x,t) >_ 0, that is, there exist, in general, characteristic roots of variable multiplicity. One has to distinguish between the case of spatial degeneracy and that of time degeneracy. The connection to the life span of solutions is given.
We prove the existence of a global in time solution of the semilinear Klein-Gordon equation in th... more We prove the existence of a global in time solution of the semilinear Klein-Gordon equation in the de Sitter space-time. 0 Introduction In this paper we prove the existence of a global in time solution of the semilinear Klein-Gordon equation in the de Sitter space-time. The metric g in the de Sitter space-time is defined as follows, g 00 = g 00 = −1, g 0j = g 0j = 0, g ij (x, t) = e 2t σ ij (x), i, j = 1, 2,. .. , n, where n j=1 σ ij (x)σ jk (x) = δ ik , and δ ij is Kronecker's delta. The metric σ ij (x) describes the time slices. In the quantum field theory the matter fields are described by a function ψ that must satisfy equations of motion. In the case of a massive scalar field, the equation of motion is the semilinear Klein-Gordon equation generated by the metric g: g ψ = m 2 ψ + V ′ ψ (x, ψ) .
High performance computations are presented for the Higgs Boson Equation in the de Sitter Spaceti... more High performance computations are presented for the Higgs Boson Equation in the de Sitter Spacetime using explicit fourth order Runge-Kutta scheme on the temporal discretization and fourth order finite difference discretization in space. In addition to the fully three space dimensional equation its one space dimensional radial solutions are also examined. The numerical code for the three space dimensional equation has been programmed in CUDA Fortran and was performed on NVIDIA Tesla K40c GPU Accelerator. The radial form of the equation was simulated in MATLAB. The numerical results demonstrate the existing theoretical result that under certain conditions bubbles form in the scalar field. We also demonstrate the known blow-up phenomena for the solutions of the semilinear Klein-Gordon equation with imaginary mass. Our numerical studies suggest several previously not known properties of the solution for which theoretical proofs do not exist yet: 1. smooth solution exists for all time if the initial conditions are compactly supported and smooth; 2. under some conditions no bubbles form; 3. solutions converge to step functions related to unforced, damped Duffing equations.
International journal of applied mathematics and statistics, 2008
We investigate the issue of the Lp − Lq estimates for the Gellerstedt operator and generalized Tr... more We investigate the issue of the Lp − Lq estimates for the Gellerstedt operator and generalized Tricomi operator in the domain where the operators are hyperbolic. A special attention is given to the case of the Cauchy problem with the initial data given outside of the set where characteristics coincide.
Nonlinear Differential Equations and Applications NoDEA
We prove the existence of global in time solution with the small initial data for the semilinear ... more We prove the existence of global in time solution with the small initial data for the semilinear equation of the spin-1 2 particles in the Friedmann-Lemaître-Robertson-Walker spacetime. Moreover, we also prove that if the initial function satisfies the Lochak-Majorana condition, then the global solution exists for arbitrary large initial value. The solution scatters to free solution for large time. The mass term is assumed to be complex-valued. The conditions on the imaginary part of mass are discussed by proving nonexistence of the global solutions if certain relation between scale function and the mass are fulfilled.
Springer Proceedings in Mathematics & Statistics, 2016
In this review, we present an integral transform that maps solutions of some class of the partial... more In this review, we present an integral transform that maps solutions of some class of the partial differential equations with time independent coefficients to solutions of more complicated equations, which have time-dependent coefficients. We illustrate this transform by applications to several model equations. In particular, we give applications to the generalized Tricomi equation, the Klein–Gordon and wave equations in the curved spacetimes such as Einstein-de Sitter, de Sitter, anti-de Sitter, and the spacetimes of the black hole embedded into de Sitter universe.
In this article we study global in time (not necessarily small) solutions of the equation for the... more In this article we study global in time (not necessarily small) solutions of the equation for the Higgs boson in the Minkowski and in the de Sitter spacetimes. We reveal some qualitative behavior of the global solutions. In particular, we formulate sufficient conditions for the existence of the zeros of global solutions in the interior of their supports, and, consequently, for the creation of the so-called bubbles, which have been studied in particle physics and inflationary cosmology. We also give some sufficient conditions for the global solution to be an oscillatory in time solution.
In this article we give sufficient and necessary conditions for the validity of the Huygens’ prin... more In this article we give sufficient and necessary conditions for the validity of the Huygens’ principle for the Dirac operator in the non-constant curvature spacetime of the Friedmann-Lemâıtre-RobertsonWalker models of cosmology. The Huygens’ principle discussed for the equation of a particle with mass m = 0 as well as a massive spin1 2 particle (field) undergoing a red shift of its wavelength as the universe expands.
Uploads
Papers by Karen Yagdjian