Uniqueness classes for solutions in unbounded domains of the first mixed problem for the

L. M. Kozhevnikova

Sbornik. Mathematics, Tome 198 (2007) no. 1, pp. 55-96

Résumé

In a cylindrical domain $D^T=(0,T)\times\Omega$, where $\Omega$ is an unbounded subdomain of $\mathbb R_{n+1}$, one considers the evolution equation $u_t=Lu$ the right-hand side of which is a quasi-elliptic operator with highest derivatives of orders $2k,2m_1,\dots,2m_n$ with respect to the variables $y_0,y_1,\dots,y_n$. For the mixed problem with Dirichlet condition at the lateral part of the boundary of $D^T$ a uniqueness class of the Täcklind type is described. For domains $\Omega$ 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 Oleǐnik 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.