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An approach to the uniqueness problem for second-order parabolic equations. (English. Russian original) Zbl 0554.35053

Sib. Math. J. 24, 694-703 (1983); translation from Sib. Mat. Zh. 24, No. 5(141), 59-70 (1983).
The Cauchy problem for the second order linear parabolic equation \((1)\quad Lu=a^{ij}(x,t)D_ iD_ ju+a^ i(x,t)D_ iu+c(x,t)u-D_ tu=0,\quad u(x,0)=u_ 0(x)\quad on\quad \Pi (T)=\{(x,t)\in R_{xt}^{n+1},\quad | x| <\infty,\quad 0<t\leq T\}\) is considered. Let S(x) be a \(C^ 2(R^ n)\) function such that a) \(S(x)\equiv 1\quad (| x| <1),\quad b)\partial /\partial rS(x)>0\quad (| x| >1),\quad c)\lim_{| x| \to \infty}S(x)=\infty.\) The authors construct U(S) a space of functions with a certain growth property dependent on S, and P(L,S), the exact class of operators L where \(a^{ij}\), \(a^ i\), c satisfy certain growth conditions as \(| x| \to \infty\). For \(L\in P(L,S)\) the uniqueness theorem for (1) in \(U(S)\cap C({\bar \Pi}(T))\cap C^{21}_{xt}(\Pi (T))\) is proved. This theorem generalizes the results of Tikhonov, Tacklind, Oleinic and Radkevich.
Reviewer: V.V.Vasil’ev

MSC:

35K15 Initial value problems for second-order parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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References:

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[2] Ya. M. Tsoraev, ?On the uniqueness classes of solutions of the first boundary-value problem in unbounded domains and of the Cauchy problem for nonuniformly parabolic equations,? Vestn. Mosk. Gos. Univ., Ser. Mat. Mekh., No. 3, 38-44 (1970). · Zbl 0216.38001
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