A Sufficient Condition for Hyperbolicity of Partial Differential Operators With Constant Coefficient Principal Part

Abstract

Let $P$ be a differential operator with principal part ${P_m}$, and suppose that ${P_m}$ has constant coefficients and is hyperbolic. It is shown that the condition for hyperbolicity of $P$ when $P$ has constant coefficients, namely, that $P$ is weaker than ${P_m}$ is also a sufficient condition for hyperbolicity in the case where $P$ does not have constant coefficients. Some generalizations are also made to the case where $P$ is a square matrix of differential operators.