Hyperbolic equations in higher dimensions

Abstract

(a) The method of spherical means The wave equation for a function u(x₁,...,x _n ,t)= u(x,t) of n space variables x₁,...,x _n and the time t is given by

$$ square u = {u_{tt}} = {c^2}\Delta u = 0 $$

(1.1)

with a positive constant c. The operator “□” defined by (1.1) is known as the D’Alembertian. For n=3 the equation can represent waves in acoustics or optics, for n=2 waves on the surface of water, for n=1 sound waves in pipes or vibrations of strings. In the initial-value problem we ask for a solution of (1.1) defined in the (n+1)-dimensional half space t>0 for which

$$ u = f\left( x \right),{\text{ }}{u_t} = g\left( x \right){\text{ for }}t = {\text{0}}. $$

(1.2)

([15], [19])