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Mat. Sb., 2007, Volume 198, Number 1, Pages 59–102 (Mi msb1519)  

This article is cited in 7 scientific papers (total in 7 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 $2k,2m_1,…,2m_n$ with respect to the variables $y_0,y_1,…,y_n$. 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.

DOI: https://doi.org/10.4213/sm1519

Full text: PDF file (878 kB)
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English version:
Sbornik: Mathematics, 2007, 198:1, 55–96

Bibliographic databases:

UDC: 517.956.4
MSC: 35K60
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 $A$”, Mat. Sb., 198:1 (2007), 59–102; Sb. Math., 198:1 (2007), 55–96

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. F. Gilimshina, “On the decay of a solution of a nonuniformly parabolic equation”, Differ. Equ., 46:2 (2010), 239–254  crossref  mathscinet  zmath  isi  elib  elib  scopus
    2. V. F. Gilimshina, F. Kh. Mukminov, “Ob ubyvanii resheniya vyrozhdayuschegosya lineinogo parabolicheskogo uravneniya”, Ufimsk. matem. zhurn., 3:4 (2011), 43–56  mathnet  zmath
    3. L. M. Kozhevnikova, “Examples of the Nonuniqueness of Solutions of the Mixed Problem for the Heat Equation in Unbounded Domains”, Math. Notes, 91:1 (2012), 58–64  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    4. V. F. Vil'danova, F. Kh. Mukminov, “Anisotropic uniqueness classes for a degenerate parabolic equation”, Sb. Math., 204:11 (2013), 1584–1597  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. Jenaliyev M.H., Amangaliyeva M., Kosmakova M., Ramazanov M., “About Dirichlet Boundary Value Problem For the Heat Equation in the Infinite Angular Domain”, Bound. Value Probl., 2014, 213  crossref  mathscinet  zmath  isi  scopus
    6. V. F. Vil'danova, F. Kh. Mukminov, “Täcklind uniqueness classes for heat equation on noncompact Riemannian manifolds”, Ufa Math. J., 7:2 (2015), 55–63  mathnet  crossref  isi  elib
    7. M. M. Amangalieva, M. T. Dzhenaliev, M. T. Kosmakova, M. I. Ramazanov, “On one homogeneous problem for the heat equation in an infinite angular domain”, Siberian Math. J., 56:6 (2015), 982–995  mathnet  crossref  crossref  mathscinet  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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