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Estimates of the Green function and a criterion for uniform stabilization of the solution of the second mixed problem for a second-order parabolic equation. (English. Russian original) Zbl 0629.35049
Sov. Math., Dokl. 31, 448-451 (1985); translation from Dokl. Akad. Nauk SSSR 282, 15-18 (1985).
The authors consider the second mixed problem in a cylindrical domain \(D=(t>0)\times \Omega\), \(\Omega \subset R^ n\), for a linear, uniformly parabolic equation of second order: \[ (1)\quad \partial u/\partial t=\sum \partial /\partial x_ i(a_{ij}(x,t)\partial u/\partial x_ j),\quad (t,x)\in D \]
\[ (2)\quad (\partial u/\partial N)|_{x\in \partial \Omega}=0,\quad u(0,x)=\phi (x)\quad \phi \in L_{\infty}(\Omega), \] where \(\partial /\partial N\) is the conormal derivative of the elliptic (for each t) operator on the right-hand side of (1). For this problem, a sharp estimate of the Green function of problem (1), (2) is deduced. A criterion for the uniform stabilization of bounded solutions of (1), (2) is established, provided \(\Omega\) satisfies some conditions allowing the following particular case: \(\Omega =\{(x_ 1,x_ 2):\) \(x_ 2>f(x_ 1)\}\subset R^ 2\), where f is an arbitrary even function, continuously differentiable on \((-\infty,+\infty)\) and of monotonously non-decreasing derivative.
Reviewer: I.Toma
35K20 Initial-boundary value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs