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On the uniform stabilization of solutions of the second mixed problem for a parabolic equation. (English. Russian original) Zbl 0554.35055
Math. USSR, Sb. 47, 439-498 (1984); translation from Mat. Sb., Nov. Ser. 119(161), No. 4(12), 451-508 (1982).
The author studies the stability of the solution of the second parabolic equation: \[ \partial u/\partial t=\sum^{n}_{i,j=1}(\partial /\partial x_ i)(a_{ij}(t,x)\partial u/\partial x_ j)=\nabla_ xA(t,x)\nabla_ xu, \] (t,x)\(\in D\), \(D=\{t>0\}\times \Omega\), \(\Omega \subset {\mathbb{R}}^ n\). A(t,x) satisfies some ellipticity and boundedness conditions.
The paper contains five sections. The first is introductory. In the second a class of uniqueness close to Täcklind classes is defined [see S. Täcklind, Nova Acta Soc. Sci. Upsal., IV Ser. 10, No. 3, 1-57 (1936; Zbl 0014.02204)]; accordingly, a class of appropriate admissible initial functions is introduced. The third section contains the proof of a fundamental inequality, satisfied by the solution. In the fourth part, some useful properties of the associated Green function are pointed out. The last section is dedicated to the stabilization of the solution; the proofs rely on the previous preparatory sections.
Reviewer: I.Toma

MSC:
35K20 Initial-boundary value problems for second-order parabolic equations
35B35 Stability in context of PDEs
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