A priori estimates for the solution of the first boundary-value problem for a class of second-order parabolic systems

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.)