A model boundary-value problem without initial conditions and with zero boundary conditions for a homogeneous Eidelman $\vec{2b}$-parabolic system of equations is considered in the domain $\{(t,x_1,\dots,x_n)\in \mathbb{R}^{n+1}|-\infty<t≤T, -∞<x_j<∞, j∈\{1,...,n-1\}, x_n>0\}$. The boundary conditions are given by differential expressions of arbitrary orders. The boundary conditions satisfy the complementarity condition of the Lopatinskii type for elliptic boundary-value problems. A proposition of the type of Liouville's theorems for analytic and harmonic functions is established for solutions of such a problem. In general, Liouville's theorems mean a statement about the determining of the form of some classes of functions with their asymptotic behavior. The proof of such theorems for solutions of the problem under consideration is based on the integral representation of the solutions and the arbitrariness of the initial hyperplane, through the values on which the representation of the solutions goes.

[1] S.D. Ivasyshen, N.I. Turchyna Green’s matrix for a model boundary-value problem with parabolic weight. Math. Methods and Physicomech. Fields 2020. 60 (4), 25–39; same: S.D. Ivasyshen, N.I. Turchyna Green’s matrix for a model boundary-value problem with parabolic weight. J. Math. Sci. 2020. 247 (1), 24–42; https:/doi.org//10.1007/s10958-020-04787-0.
[2] Turchyna N. I., Ivasyshen S. D. On the model boundary value problem with vector weight. Bukovinian Math. J. 2017. 5 (3–4), 63–167.
[3] Turchyna N. I., Ivasyshen S. D. Correct solvability of a model−→2b-parabolic boundary-value problem in Holder spaces. Bukovinian Math. J. 2018. 6 (3–4), 152–164; https:// doi.org/10.31861/bmj2018.03.152.
[4] Turchyna N. I., Ivasyshen S. D. Correct solvability in Holder spaces of growing functions of model boundary-value problems with and without initial conditions for a system parabolic in the sense of Eidelman. Math. Methods and Physicomech. Fields. 2019. 62 (2), 7–25.
[5] Turchyna N. I., Ivasyshen S. D. On integral representation of the solutions of a model −→2b-parabolic boundary value problem. Carpathian Math. Publ. 2019. 11 (1), 193–203.