Geodesic
A priori estimates for the solution of the first boundary-value problem for a class of second-order parabolic systems
Izvestiya. Mathematics, Tome 65 (2001) no. 4, pp. 705-726 
L. I. Kamynin; B. N. Khimchenko. A priori estimates for the solution of the first boundary-value problem for a class of second-order parabolic systems. Izvestiya. Mathematics, Tome 65 (2001) no. 4, pp. 705-726. http://geodesic.mathdoc.fr/item/IM2_2001_65_4_a4/
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We consider two classes of second-order parabolic matrix-vector systems (with solutions $u\in M_{m\times 1}$, $m\geqslant 2$) that can be reduced to a single second-order parabolic equation for a scalar function $v=\langle p,u\rangle$, where $p\in M_{m\times 1}$ is a fixed stochastic constant vector. We consider the first boundary-value problem for a scalar second-order parabolic equation (with unbounded coefficients) in a domain unbounded with respect to $x$ under the assumption of strong absorption at infinity. We obtain an a priori estimate for solutions of the first boundary-value problem in the generalized Tikhonov–Täcklind classes. (The problem under investigation has at most one solution in these classes.)

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