Uniqueness classes for solutions in unbounded domains of the first mixed problem for the equation with quasi-elliptic operator

Abstract

In a cylindrical domain , where is an unbounded subdomain of , one considers the evolution equation the right-hand side of which is a quasi-elliptic operator with highest derivatives of orders with respect to the variables . For the mixed problem with Dirichlet condition at the lateral part of the boundary of a uniqueness class of the Täcklind type is described. For domains tapering at infinity another uniqueness class is distinguished, a geometric one, which is broader than the Täcklind-type class. It is shown that for domains with irregular behaviour of the boundary this class is wider than the one described for a second-order parabolic equation by Oleĭnik and Iosif'yan (Uspekhi Mat. Nauk, 1976 [17]). In a wide class of tapering domains non-uniqueness examples for solutions of the first mixed problem for the heat equation are constructed, which supports the exactness of the geometric uniqueness class. Bibliography: 33 titles.