Linear degenerate parabolic equations of arbitrary order with a finite region of dependence

The Cauchy problem is considered for equations of the form $u_l-Lu=0$, where $Lu=L(i,x_1,\dots,x_n,\partial/\partial x_1,\dots,\partial x_n)u$ is an elliptic differential expression of arbitrary order which is degenerate for certain values of the arguments in the first order differential expression. Conditions are stated on the nature of the degeneracy which are sufficient for a solution of this problem to have a finite region of dependence.