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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)
##### Keywords:
uniqueness; dissipativity
Full Text:
##### References:
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