On Global Solutions of Two-Dimensional Hyperbolic Equations with General-Kind Nonlocal Potentials

Abstract

In the case of one spatial independent variable, we study hyperbolic differential-difference equations with potentials represented as linear combinations of translations of the desired function along the spatial variable. The qualitative novelty of this investigation is that, unlike previous research, it is not assumed that the real part of the symbol of the differential-difference operator contained in the equation has a constant sign. Previously, it was possible to remove that substantial restriction (i.e., the specified sign constancy) only for the case where the nonlocal term (i.e., the translated potential) is unique. In the present paper, we consider the case of the general-kind one-variable nonlocal potential, i.e., equations with an arbitrary amount of translated terms. No commensurability assumptions are imposed on the translation lengths. The following results are presented: We find a condition relating the coefficients at the nonlocal terms of the investigated equation and the length of the translations, providing the global solvability of the investigated equation. Under this condition, we explicitly construct a three-parametric family of smooth global solutions of the investigated equation.

Keywords

• differential-difference operators
• hyperbolic equations
• nonlocal potentials
• smooth solutions
• incommensurability of translations