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Dissipative effect for second-order parabolic operators. (English. Russian original) Zbl 0677.35049
Sib. Math. J. 29, No. 5, 791-800 (1988); translation from Sib. Mat. Zh. 29, No. 5(171), 131-142 (1988).
The principal result of this paper is a uniqueness theorem for the problem \[ (6)\quad a^{ij}(x,t)D_ iD_ ju+b^ iD_ iu+c(x,t)u-D_ tu=0 \] ((x,t) in the domain \(\pi (0,\infty)=\{(x,t):\) \(| x| <\infty\), \(0<t<\infty \}),\)
(7) \(u(x,0)=0\), in the class \(C(\pi [0,\infty)\wedge C^{2,1}_{x,t}(\pi (0,\infty)),\)
where: \((a^{ij}(x,t))\) is a real symmetric matrix with eigenvalues \(\lambda^ 1(x,t);...;\lambda^ n(x,t)\), \(0\leq \lambda^ 1\leq...\leq \lambda^ n\), \(b^ i\) \((i=1,...,n)\) and c(x,t) are local bounded functions in \(\pi\) (0,\(\infty).\)
If there exist two functions \(p\in P(p)\) and \(q\in T_ p(g)\) (P(p) and \(T_ p(g)\) are some particular subsets of C([0,\(\infty)))\) and \(u_ 0\in C([0,\infty))\) so that \[ 2\lambda^ 1(x,t)(3+q)p^ 2G(| x|)\leq -\rho^ 2(| x|)c(x,t), \] on \(\pi\) (0,\(\infty)\), for \(u=1\) (here \(q=const>0\), \(G(s)=\int^{s}_{0}(1/g(z))dz\), \(s\in (0,\infty))\) \[ 2\lambda^ n(x,t)\max [(\beta +q)p^ 2(G(| x|)),2(n-1)p(G(| x|))g(| x|)/| x| \leq -g^ 2(| x|)c(x,t)], \] on \(\pi\) (0,\(\infty)\), for \(n\geq 2\), \[ 4b(x,t)p(G(| x|))\leq g^ 2(| x|)c(x,t) \] on \(\pi\) (0,\(\infty)\) (if a dissipativity condition is given) and also \[ | u(x,t)| \leq u_ 0(t)\exp \{\int^{G(| x|)}_{0}p(s)ds\} \] on \(\pi\) (0,\(\infty)\), then u(x,t)\(\equiv 0\) on \(\pi\) [0,\(\infty)\).
Reviewer: I.Onciulescu

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