The Dependence of Solution Uniqueness Classes of Boundary Value Problems for General Parabolic Systems on the Geometry of an Unbounded Domain

General boundary value problems are considered for general parabolic (in the Douglas–Nirenberg–Solonnikov sense) systems. The dependence of solution uniqueness classes of these problems on the geometry of a nonbounded domain is established. The dependence of solution uniqueness classes of the first boundary value problem for a second-order parabolic equation in an unbounded domain on the domain geometry was considered in [2]. It was established there that the uniqueness class could be wider than the solution uniqueness class of the Cauchy problem for the above-mentioned equation. Analogous results were obtained in [2] by the method of barrier functions. In [3] O. Oleinik constructed examples of second-order parabolic equations in the exponentially narrowing domains for which the solution uniqueness class for the second boundary value problem is the same as the solution uniqueness class of the Cauchy problem although the solution uniqueness class in this domain is wider. Later, in [4] E. Landis showed that if the domain narrows with a sufficient quickness at |x| → ∞, then the solution uniqueness class for the second boundary value problem can be wider than that of the Cauchy problem. In [5] the author considered degenerating parabolic equations of second order and obtained analogous uniqueness theorems for general boundary value problems. This paper deals with general boundary value problems for general parabolic systems in unbounded domains. Such problems were studied in [6], where solvability conditions similar to the Shapiro-Lopatinski conditions for 1991 Mathematics Subject Classification. 35K50.