The Klein-Gordon equation with light-cone data

Abstract

It is shown that the characteristic Cauchy problem$\left( {\frac{{\partial ^2 }}{{\partial t^2 }} - \Delta + 1} \right)$·u(x,t)=0,u(x,−|x|)=f(x),x∈ℝⁿ,n≧1 has a unique finite energy weak solution for allf such that ∫dx(|∇f|²+|f|²)<∞ and all finite energy weak solutions of the equation are obtained in this way.