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This article is cited in 8 scientific papers (total in 8 papers)
Uniqueness classes for solutions in unbounded domains of the first mixed problem for the
equation $u_t=Au$ with quasi-elliptic operator $A$
L. M. Kozhevnikova Sterlitamak State Pedagogical Institute
Abstract:
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.
Received: 30.01.2006 and 31.08.2006
Citation:
L. M. Kozhevnikova, “Uniqueness classes for solutions in unbounded domains of the first mixed problem for the
equation $u_t=Au$ with quasi-elliptic operator $A$”, Sb. Math., 198:1 (2007), 55–96
Linking options:
https://www.mathnet.ru/eng/sm1519https://doi.org/10.1070/SM2007v198n01ABEH003829 https://www.mathnet.ru/eng/sm/v198/i1/p59
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