On solutions of a parabolic equation that decrease with respect to the space variables

The equation $L(x,D)u(x)=0$ is considered, where the operator $L(x,D)$ in the region is representable in the form $L(x,D)=L_m(x,D)+L_0(x, D)$; here $L_m(x,D)$ has order $m$, real coefficients in $C^1$, contains derivatives in the variables $x_1,\dots,x_k$, $k<n$, and is elliptic in these variables, and for any real vector $N=\{N_1,\dots,N_k\}\ne0$ the equation $L_m(x,\xi+i\tau N)=0$, $\xi=\{\xi_1,\dots,\xi_k\}$, for any real $\xi$ not proportional to $N$ does not have double real zeros $\tau$.
Bibliography: 3 titles.