Karen Yagdjian

Global Existence for the n-Dimensional Semilinear Tricomi-Type Equations

In this article we investigate the issue of global existence of the solutions of the Cauchy problem for semilinear Tricomi-type equations in ℝ n+1, n > 1. We give some sufficient conditions for existence of the global weak solutions. These conditions tie together nonlinearity with the speed of propagation and with the dimension n. We also prove necessity of these (or close) conditions. In fact, we extend these necessity results to the nonlocal semilinear equations.

Global Existence in the Cauchy Problem for Nonlinear Wave Equations with Variable Speed of Propagation

We consider nonlinear wave equations with variable coefficients. Special attention is devoted to the parametric resonance phenomena.

A note on wave equation in Einstein and de Sitter space-time

We consider the wave propagating in the Einstein and de Sitter space-time. The covariant d’Alembert’s operator in the Einstein and de Sitter space-time belongs to the family of the non-Fuchsian partial differential operators. We introduce the initial value problem for this equation and give the explicit representation formulas for the solutions. We also show the Lp−Lq estimates for solutions.

On the Global Solutions of the Higgs Boson Equation

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 oscillatory in time.

Semilinear Hyperbolic Equations in Curved Spacetime

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.

Global Solutions of Semilinear System of Klein-Gordon Equations in de Sitter Spacetime

In this article we prove global existence of small data solutions of the Cauchy problem for a system of semilinear Klein-Gordon equations in the de Sitter spacetime. The mass matrix is assumed to be diagonalizable with positive eigenvalues. The existence is proved under the assumption that the eigenvalues are outside of some open bounded interval.

H∞ Well-Posedness for Degenerate p-Evolution Models of Higher Order with Time-Dependent Coefficients

In this paper we deal with time dependent p-evolution Cauchy problems. The differential operators have characteristics of variable multiplicity. We consider a degeneracy only in t=0. We shall prove a well-posedness result in the scale of Sobolev spaces using a C 1-approach. In this way we will prove H ∞ well-posedness with an (at most) finite loss of regularity.

Integral Transform Approach to Time-Dependent Partial Differential Equations

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.

Integral Transform Approach to Solving Klein–Gordon Equation with Variable Coefficients

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 model equations. In particular, we give applications to the Klein–Gordon and wave equations in the curved spacetimes such as the de Sitter universe.

Integral Transform Approach to the Cauchy Problem for the Evolution Equations

In this note we describe some integral transform that allows to write solutions of the Cauchy problem for one partial differential equation via solution of another one. It was suggested by author in J. Differ. Equ. 206:227–252, 2004 in the case when the last equation is a wave equation, and then used in the series of articles (see, e.g., Yagdjian in J. Differ. Equ. 206:227–252, 2004, Yagdjian and Galstian in J. Math. Anal. Appl. 346(2):501–520, 2008, Yagdjian and Galstian in Commun. Math. Phys. 285:293–344, 2009, Yagdjian in Rend. Ist. Mat. Univ. Trieste 42:221–243, 2010, Yagdjian in J. Math. Anal. Appl. 396(1):323–344, 2012, Yagdjian in Commun. Partial Differ. Equ. 37(3):447–478, 2012, Yagdjian in Semilinear Hyperbolic Equations in Curved Spacetimepp, pp. 391–415, 2014 and Yagdjian in J. Math. Phys. 54(9):091503, 2013) to investigate several well-known equations such as Tricomi-type equation, the Klein–Gordon equation in the de Sitter and Einstein–de Sitter spacetimes. The generalization given in this note allows us to consider also evolution equations with x-dependent coefficients.