On the uniqueness of solutions of a certain characteristic Cauchy problem

Abstract

The question of uniqueness of solutions of the global Cauchy problem (1)–(2) below is discussed. We assume that there exists a complex constant c such that the modified equation $$\frac{{\partial ^2 u}}{{\partial t^2 }} = c_{\left| \alpha \right| \leqq 2} \sum a_\alpha (x) D_x^\alpha u$$ becomes hyperbolic. Under this and some other additional conditions (See Condition A in §2) we prove the uniqueness of solutions of the Cauchy problem within the class of functions u(t, x) such that $$|u(t,x)| \leqq C exp(a|x|^2 ) ,$$ C and a being positive constants