Some Remarks on the Dependence Domain for Weakly Hyperbolic Equations with Constant Multiplicity

Abstract

We are concerned with the Cauchy problem for the following weakly hyperbolic operator with constant multiplicity:

$$\matrix{ {P(x,t;{D_x},{D_t})u(x,t) = f(x,t),\;(x,t) \in {{\bf{R}}^n} \times [{t_0},T]} \cr {\partial _t^ju{|_{t = {t_0}}} = {u_j}(x)\;(0 \le j \le m - 1),\;{t_0} \in [0,T],} \cr }$$

((1.1))

where D _x = i ⁻¹∂/∂x, D _t = i ⁻¹∂/∂t. Hereafter we write D instead of D _x . We denote Ω = R ⁿ × [0, T].