Lp−LqL^p-L^q-Decay Estimates for the Klein-Gordon Equation in the Anti-de~Sitter Space-Time

We derive $L^p-L^q$ - decay estimates for the solutions of the Cauchy problem for the Klein-Gordon equation in the anti-de Sitter spacetime, that is, for $\Box_g u - m^2 u = f$ in models of mathematical cosmology. The obtained $L^p$--$L^q$ estimates imply exponential decay of the solutions for large times

Finite lifespan of solutions of the semilinear wave equation in the Einstein–de Sitter spacetime

We examine the solutions of the semilinear wave equation, and, in particular, of the φq model of quantum field theory in the curved space-time. More exactly, for 1 \u3c q \u3c 4 we prove that the solution of the massless self-interacting scalar field equation in the Einstein-de Sitter universe has finite lifespan

Fundamental solutions for the Dirac equation in curved spacetime and generalized Euler-Poisson-Darboux equation

We present the fundamental solutions for the spin-1/2 fields propagating in spacetimes with power type expansion/contraction and the fundamental solution of the Cauchy problem for the Dirac equation. The derivation of these fundamental solutions is based on formulas for the solutions to the generalized Euler-Poisson-Darboux equation, which are obtained by the integral transform approach

The global existence of small self-interacting scalar field propagating in the contracting universe

We present a condition on the self-interaction term that guaranties the existence of the global in time solution of the Cauchy problem for the semilinear Klein-Gordon equation in the Friedmann-Lamaˆitre-Robertson-Walker model of the contracting universe. For the Klein- Gordon equation with the Higgs potential we give a lower estimate for the lifespan of solution

A Note on Wave Equation in Einstein & de Sitter Spacetime

We consider the wave propagating in the Einstein & de Sitter spacetime. The covariant d'Alembert's operator in the Einstein & de Sitter spacetime 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 $L^p - L^q$ estimates for solutions

Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime

In this article we construct the fundamental solutions for the Klein-Gordon equation in de Sitter spacetime. We use these fundamental solutions to represent solutions of the Cauchy problem and to prove $L^p-L^q$ estimates for the solutions of the equation with and without a source term

On deconvolution methods

Several methods for solving efficiently the one-dimensional deconvolution problem are proposed. The problem is to solve the Volterra equation ku:=R t 0 k(t \Gamma s)u(s)ds = g(t); 0 ^ t ^ T. The data, g(t), are noisy. Of special practical interest is the case when the data are noisy and known at a discrete set of times. A general approach to the deconvolution problem is proposed: represent k = A(I + S), where a method for a stable inversion of A is known, S is a compact operator, and I + S is injective. This method is illustrated by examples: smooth kernels k(t), and weakly singular kernels, corresponding to Abel-type of integral equations, are considered. A recursive estimation scheme for solving deconvolution problem with noisy discrete data is justified mathematically, its convergence is proved, and error estimates are obtained for the proposed deconvolution method