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Existence of a solution to the scattering problem for the ultrahyperbolic equation

Maxim N. Demchenko

Abstract

We consider the ultrahyperbolic equation in the Euclidean space. The behavior at the infinity of a certain class of solutions is studied. We examine the issue of existence of solutions to the scattering problem: for a given asymptotics at the infinity, the corresponding solution to the equation is constructed.

Existence of a solution to the scattering problem for the ultrahyperbolic equation

Abstract

We consider the ultrahyperbolic equation in the Euclidean space. The behavior at the infinity of a certain class of solutions is studied. We examine the issue of existence of solutions to the scattering problem: for a given asymptotics at the infinity, the corresponding solution to the equation is constructed.

Paper Structure

This paper contains 4 sections, 7 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. A family of solutions exhibiting asymptotics (\ref{['scat-data']})
  3. Asymptotic behavior of solutions of the form (\ref{['hatu-f']})
  4. Auxiliary assertions

Key Result

Theorem 1

Let a function $f(\theta, \omega, p)$ on $S^{d-1}\times S^{n-1}\times{\mathbb R}$ satisfy the following conditions for all $k\geqslant 1$, $|\alpha|\geqslant 0$, and some $0<\varepsilon\leqslant 1/2$, and let the following relation hold: Then there exists a $C^\infty$-smooth solution $u(x,y)$ to equation (eqn) satisfying (scat-data).

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 2 more