Properties of solutions of linear evolutionary systems with elliptic space part

We study the system $\mathscr L(t,x;\frac\partial{\partial t},D_x)u=f$, where $\mathscr L$ is an $N\times N$ matrix such that the matrix $\mathscr L(t,x;0,i,\sigma)$ is uniformly Petrovskii elliptic. We establish unimprovable estimates of the growth of the solutio belonging to a convex cone of the space $C^N$ in a band, in a halfspace, and in the entire space. These estimates are applied to obtain new uniqueness theorems for Cauchy's problem.
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